Stable homotopy over the Steenrod algebra John H. Palmieri Author address: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556 E-mail address: palmieri@member.ams.org 1991 Mathematics Subject Classification. 55S10, 55U15, 18G35, 55U35, 55T15, 55P42, 55Q10, 55Q45, 18G15, 16W30, 18E30, 20J99. Research partially supported by National Science Foundation grant DMS-9407459. Abstract.We apply the tools of stable homotopy theory to the study of mod- ules over the Steenrod algebra A*; in particular, we study the (triangula* *ted) category Stable(A) of unbounded cochain complexes of injective comodules over A, the dual of A*, in which the morphisms are cochain homotopy class* *es of maps. This category satisfies the axioms of a stable homotopy category* * (as given in [HPS97]); so we can use Brown representability, Bousfield locali* *za- tion, Brown-Comenetz duality, and other homotopy-theoretic tools to study Ext**A(Fp; Fp), which plays the role of the stable homotopy groups of sph* *eres. We also have nilpotence theorems, periodicity theorems, a convergent chro- matic tower, and a number of other results. Contents List of Figures v Preface vii Chapter 1. Stable homotopy over a Hopf algebra 1 1.1. Recollections 2 1.1.1. Hopf algebras 2 1.1.2. Comodules 4 1.1.3. Homological algebra 5 1.2. The category Stable() 6 1.3. The functor H 8 1.3.1. Remarks on Hopf algebra extensions 10 1.4. Some classical homotopy theory 13 1.5. The Adams spectral sequence 15 1.6. Bousfield classes and Brown-Comenetz duality 18 1.7. Further discussion 19 Chapter 2. Basic properties of the Steenrod algebra 23 2.1. Quotient Hopf algebras of A 23 2.1.1. Quasi-elementary quotients of A 28 2.2. Pst-homology 29 2.3. Vanishing lines for homotopy groups 34 2.3.1. Proof of Theorems 2.3.1 and 2.3.2 for p = 2 35 2.3.2. Changes necessary when p is odd 40 2.4. Self-maps via vanishing lines 42 2.5. Further discussion 44 Chapter 3. Quillen stratification and nilpotence 47 3.1. Statements of theorems 48 3.1.1. Quillen stratification 48 3.1.2. Nilpotence 50 3.2. Nilpotence and F-isomorphism via the Hopf algebra D 51 3.2.1. Nilpotence: Proof of Theorem 3.1.5 54 3.2.2. F-isomorphism: Proof of Theorem 3.1.2 55 3.3. Nilpotence and F-isomorphism via quasi-elementary quotients 56 3.3.1. Nilpotence: Proof of Theorem 3.1.6 56 3.3.2. F-isomorphism: Proof of Theorem 3.1.3 58 3.4. Further discussion: nilpotence at odd primes 60 3.5. Further discussion: miscellany 61 iii iv CONTENTS Chapter 4. Periodicity and other applications of the nilpotence theorems 63 4.1. The periodicity theorem 63 4.2. Properties of y-maps 65 4.3. The proof of the periodicity theorem 67 4.4. Computation of some invariants in HD** 69 4.5. Computation of a few Bousfield classes 73 4.6. Ideals and thick subcategories 76 4.6.1. Ideals 76 4.6.2. A thick subcategory conjecture 78 4.7. Construction of spectra of specified type 80 4.8. Further discussion: slope supports 84 4.9. Further discussion: miscellany 86 Chapter 5. Chromatic structure 87 5.1. Margolis' killing construction 87 5.2. A Tate version of the functor H 94 5.3. Chromatic convergence 97 5.4. Further discussion 99 Appendix A. Two technical results 101 A.1. An underlying model category 101 A.2. Vanishing planes in Adams spectral sequences 102 A.2.1. Vanishing lines in ordinary stable homotopy 107 Appendix B. Steenrod operations and nilpotence in Ext**(k; k) 109 B.1. Steenrod operations in Hopf algebra cohomology 109 B.2. Nilpotence in HB**= Ext**B(F2; F2) 110 B.3. Nilpotence in HB**= Ext**B(Fp; Fp) when p is odd 111 B.3.1. Sketch of proof of Conjecture B.3.4, and other results 113 Bibliography 117 Index 121 List of Figures 2.1.A Graphical representation of a quotient Hopf algebra of A. 25 2.1.B Profile function for A(n). 26 2.1.C Profile functions for maximal elementary quotients of A, p = 2. 28 2.3.A Vanishing line at the prime 2. 34 3.1.A Profile function for D. 48 3.2.B Profile function for D(n). 52 3.3.C Profile functions for Dr and Dr;q. 57 4.4.A Graphical depiction of coaction of A on limHE** 72 4.8.A T(t; s) and T(m) as subsets of Slopes0. 85 5.1.A Vanishing curve for ssij(CfnS0). 93 5.2.A The coefficients of HA(1)and bHA(1). 96 v vi LIST OF FIGURES Preface The object of study for this book is the mod p Steenrod algebra A and its co* *ho- mology ExtA. Various people (including the author) have approached this subject by taking results in stable homotopy theory and then trying to prove analogous results for A-modules. This has proven to be successful, but the analogies were* * just that_there was no formal setting in which to do anything more precise than to make analogies. In [HPS97 ], Hovey, Strickland, and the author developed "axiomatic stable homotopy theory." In particular, we gave axioms for a stable homotopy category;* * in any such category, one has available many of the tools of classical and modern * *stable homotopy theory_tools like Brown representability and Bousfield localization. It turns out that a category Stable(A) (defined in the next paragraph) related to * *the category of A-modules is such a category; as one might expect, the trivial modu* *le Fp plays the role of the sphere spectrum S0, and Ext**A(- ; -) plays the role of homotopy classes of maps. Since so many of the tools of stable homotopy theory are focused on the study of the homotopy groups of S0 (and of other spectra), one should expect the corresponding tools in Stable(A) to help in the study of Ext**A(Fp; Fp) (and related groups). In this book we apply some of these tools (nilpotence theorems, periodicity theorems, chromatic towers, etc.) to the stud* *y of Ext over the Steenrod algebra. It is our hope that this book will serve two pur* *poses: first, to provide a reference source for a number of results about the cohomolo* *gy of the Steenrod algebra, and second, to provide an example of an in-depth use of t* *he language and tools of axiomatic stable homotopy theory in an algebraic setting. First we describe the category in which we work. We fix a prime p, let A* be the mod p Steenrod algebra, and let A = Hom Fp(A*; Fp) be the (graded) dual of the Steenrod algebra. We let Stable(A) be the category whose objects are cochain complexes of injective left A-comodules, and whose morphisms are cochain homo- topy classes of maps. This is a stable homotopy category (of a particularly nice sort_it is a monogenic Brown category_see [HPS97 , 9.5]). We prove a number of results in Stable(A); some of these are analogues of results in the ordinary* * stable homotopy category, and some are not. Some of these are new, and some already known, at least in the setting of A*-modules; the old results often need new pr* *oofs to apply in the more general setting we discuss here. Note. This work arose from the study of the abelian category of (left) A*- modules; to apply stable homotopy theoretic techniques, though, it is most con- venient to work in a triangulated category. One's first guess for an appropria* *te category might have objects which are chain complexes of projective A*-modules; it turns out that this category has some technical difficulties (see Remark 1.2* *.1). It is much more convenient to work with A-comodules instead of A*-modules, and fortunately, one does not lose much by doing this. Most A*-modules of interest * *can vii viii PREFACE be viewed as A-comodules; the main effects of using comodules are things of the following sort: various arrows go the "wrong" way, Ext**A(k; k) is covariant in* * A, and one studies A by means of its quotient Hopf algebras (because those are dual to the sub-Hopf algebras of A*). Each chapter is divided into a number of sections; at the beginning of each chapter, we give a brief description of its contents, section by section. In t* *his introduction, we give a brief overview of each chapter. We note that each chapt* *er has at least one "Further discussion" section, in which we discuss issues auxil* *iary to the general discussion. In Chapter 1, we set up notation and discuss results that hold in the catego* *ry Stable() for any graded commutative Hopf algebra over a field k, e.g., the dual of a group algebra, the dual of an enveloping algebra, or the dual of the Steen* *rod algebra. Aside from setting up notation for use throughout this book, the main topics of this chapter include: construction of cellular and Postnikov towers, * *an examination of the Adams spectral sequence associated to particular homology theories on Stable(), and some remarks on Bousfield classes and Brown-Comenetz duality. Note. While some of Chapter 1 may be well-known, we recommend that the reader look over Section 1.2 and the first part of Section 1.3 (at least the de* *finition of the functor H) before reading later parts of the book. These sections introd* *uce notation that gets used throughout the book. In Chapter 2 we specialize to the case in which p is a prime, k = Fpis the f* *ield with p elements, and A is the dual of the mod p Steenrod algebra. Recall from [Mil58] that as algebras, we have ( A ~= F2[1; 2; 3; : :]:; if p = 2, Fp[1; 2; 3; : :]: E[o0; o1; o2;i:f:]:;p is odd. The coproduct on A is determined by Xn i : n 7-! pn-i i; i=0 Xn i : on 7-! pn-i oi+ on 1: i=0 In this chapter, we discuss two tools with which to study Stable(A): quotient H* *opf algebras of A and Pst-homology. We use these tools to prove two theorems: the first is a vanishing line theorem (given conditions on the Pst-homology groups * *of X, then Exts;tA(Fp; X) = 0 when s > mt-c, for some numbers m and c). The second is a "self-map" theorem: given a finite A-comodule M, we construct a non-nilpotent element of Ext**A(M; M) satisfying certain properties. Let p = 2. In Chapter 3 we develop analogues in the category Stable(A) of the nilpotence theorem of Devinatz, Hopkins, and Smith, as well as the stratificati* *on theorem of Quillen. In fact we give two nilpotence theorems: in one we describe a single ring object (like BP) that detects nilpotence; more precisely, there i* *s a quotient Hopf algebra D of A so that, if M is a finite-dimensional A-comodule, an element z 2 Ext**A(M; M) is nilpotent under Yoneda composition if and only if its restriction to Ext**D(M; M) is nilpotent. The second nilpotence theorem* * is similar, but uses a family of ring objects (somewhat like the Morava K-theories) to detect nilpotence. These are versions in Stable(A) of the nilpotence theorems PREFACE ix of [DHS88 ] and [HSb ]. We strengthen these results when studying Ext**A(F2; F* *2), by "identifying" the image of Ext**A(F2; F2) -!Ext**D(F2; F2) (and similarly fo* *r the other nilpotence theorem). One can view this as an analogue of Quillen's theorem [Qui71 , 6.2], in which he identifies the cohomology of a compact Lie group up * *to F-isomorphism. Again, let p = 2. In Chapter 4, we discuss applications of the theorems from the previous chapter. In ordinary stable homotopy theory, the nilpotence theore* *ms lead to the periodicity theorem and the thick subcategory theorem (see [Hop87 ]* *); in our setting, things are a bit harder, so we get a weak version of a periodic* *ity theorem, and only a conjecture as to a classification of the thick subcategorie* *s of finite objects in Stable(A). More precisely, if M is a finite-dimensional A-com* *odule, then we produce a number of central non-nilpotent elements in Ext**A(M; M) by using the "variety of M": the kernel of Ext**D(F2; F2) -!Ext**D(M; M). One of our analogues of Quillen's theorem says that the elements of the kern* *el of Ext**A(F2; F2) -! Ext**D(F2; F2) are nilpotent, and it identifies the image.* * This identification is not explicit, so we discuss a small list of examples. We also* * imi- tate [Rav84 ] to show that the objects that detect nilpotence have strictly sma* *ller Bousfield class than the sphere. Let p be any prime. In Chapter 5 we consider Steenrod algebra analogues of chromatic theory and the functors Ln and Lfn. The latter turns out be more tractable; in fact, it is a generalization (from the module setting) of Margoli* *s' killing construction [Mar83 , Chapter 21]. We show that Ln 6= Lfnif n > 1, at least at the prime 2. We compute Lfnon some particular ring spectra, and show that, at least for these rings, it turns "group cohomology" into "Tate cohomolo* *gy." We use this result to show that the chromatic tower constructed using the funct* *ors Lfnconverges for any finite object. (This is an extension of a theorem of Margo* *lis [Mar83 , Theorem 22.1].) We also have several appendices: In Appendix A.1, we describe a model cat- egory whose associated homotopy category is Stable(A); the results in this sect* *ion are due to Hovey. In Appendix A.2 we prove a theorem due to Hopkins and Smith [HSa ], that the property of having a vanishing line with given slope, at some * *term of the Adams spectral sequence and with some intercept, is generic. (We prove t* *his in the context of Adams spectral sequences in Stable(A), which are trigraded; h* *ence we actually discuss vanishing planes.) In Appendix B, we discuss the nilpotence* * of certain classes in Ext**A(Fp; Fp) when A is the Steenrod algebra; we use these * *results in Chapter 3 to prove our nilpotence theorems. In this book we have a mix of results: some are extensions of older results to the cochain complex setting, and some are new. For each older result, if the proof in the literature extends easily to our setting, then we do not include a proof; otherwise, we at least give a sketch. It appears that when one uses the language of stable homotopy theory, one tends to change arguments with spectral sequences into simpler arguments with cofibration sequences (see Lemma 1.3.10, for example), so even though the setting is potentially more complicated, some * *of the proofs simplify. In such cases, we often give in to temptation and include * *the new proof in its entirety (as, for example, with the vanishing line theorem 2.3* *.1). Obviously, we include full proofs of all of the new results, and we give comple* *te references for all of the old results. x PREFACE Acknowledgments: I have had a number of entertaining and illuminating dis- cussions with Mark Hovey, Mike Hopkins, and Haynes Miller on this material. CHAPTER 1 Stable homotopy over a Hopf algebra In this chapter we discuss stable homotopy theory over any graded commutative Hopf algebra over a field k; a major focus of study is Ext**(k; k), where Ext denotes comodule Ext_derived functors of Hom in the category of left A-comodule* *s. This material applies when is the dual of a group algebra, the dual of an enve* *loping algebra, or the dual of the Steenrod algebra; in these cases, Ext**(k; k) is the ordinary cohomology of * with coefficients in k. Our goal in this chapter is to establish some notation, make some basic def- initions, and prove some general facts about the category Stable() of cochain complexes of injective -comodules. In more detail: we start in Section 1.1 with brief recollections about Hopf * *al- gebras, comodules, and homological algebra. In Section 1.2 we define our setting for the rest of the book, the category Stable(). We also set up the some im- portant notation; for instance, we explain the grading conventions on morphisms and (co)homology functors in Stable()_if X is an injective resolution of a left comodule M, then the (s; t)-homotopy group sss;tX is equal to Exts;t(k; M). In Section 1.3 we construct some particular ring objects in our category, one obje* *ct HB for every quotient Hopf algebra B of . To be precise, HB is an injective res- olution of 2Bk; this is a ring spectrum in Stable(). So for instance, if we we* *re working with = (kG)*, we would have one such object HB for every subgroup B of G, and the object HB would have homotopy groups ss*HB = H*(B; k). In Subsection 1.3.1 we establish some notation for Hopf algebra extensions, and we prove one or two useful results about extensions with small kernel. For example, given an extension of Hopf algebras of the form E[x] -!B -!C; the associated extension spectral sequence has only one possible differential; * *if B is a quotient Hopf algebra of , then in the category Stable(), this manifests itse* *lf as a cofibration sequence HB -!HB -!HC. In Section 1.4 we set up cellular towers and Postnikov towers in Stable(), and we prove a Hurewicz theorem and a few useful lemmas. In Section 1.5 we discuss the Adams spectral sequence based on t* *he homology theory associated to the ring spectrum HB, for B a "conormal" quotient of . This turns out to be the same, up to a regrading, as the Lyndon-Hochschild- Serre spectral sequence associated to the Hopf algebra extension 2Bk -! -!B: In Section 1.6 we define Bousfield classes and Brown-Comenetz duality, and we recall some results of Ravenel's relating the two. In Section 1.7 we apply some of this work to the study of stable homotopy ov* *er a group algebra. We point, for example, out that a corollary of work of Benson, 1 2 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA Carlson, and Rickard is a classification of the Bousfield lattice in Stable(kG** *), for G a p-group and k a field of characteristic p. 1.1.Recollections In this section we review material on Hopf algebras, comodules, and homologi* *cal algebra. This material is standard; many readers are probably familiar with it. 1.1.1. Hopf algebras. We start with the definition of a Hopf algebra. This is standard; one reference is [MM65 ]. DefinitionL1.1.1.Fix a field k. A Hopf algebra over k is a graded k-vector space = i2Zitogether with the following structure maps: oa unit map j :k -!, oa multiplication map : k -!, oa counit map ": -!k, oa comultiplication map : -! k , oand a conjugation map O: -!. The maps j and give the structure of an associative unital k-algebra_i.e., the following two diagrams commute (all tensor products are over k): -1---! ? ? 1 ?y ?y ----! ; and k[ | | [ = 1j | [ | [] |u u _____:w | | aeaeo j1 | = | ae |ae k Dually, the maps " and make into a coassociative coalgebra_the dual diagrams to those above (i.e., diagrams as above, but with all the arrows reversed) comm* *ute. We give an algebra structure via the composite 1T1----! ---! ; where T : -! is the "twist" map: T(a b) = (-1)|a||b|b a. (Dually, we can give the structure of a coalgebra.) We insist that the maps and " be algebra maps; equivalently, we insist that and j be coalgebra maps. Lastly, 1.1. RECOLLECTIONS 3 the conjugation map O makes the following diagram commute: 1O _______w _____w | | | | " | | | | | | |u j |u k ___________________:w Also, the same diagram, except with O 1 replacing 1 O, also commutes. Convention 1.1.2.We work throughout with graded vector spaces; every map between graded spaces is a graded map, and every element from such a vector spa* *ce is assumed to be homogeneous, unless otherwise indicated. Given v 2 V , we write |v| for the degree of v. Also, all unmarked tensor products are over the ground* * field. Since the structure maps in the previous definition are assumedLto be graded maps, then the image of j lies in 0, and the kernel of " contains i6=0i. Definition 1.1.3.Let be a Hopf algebra over a field k. We say that is commutative if it is commutative as an algebra_i.e., the following diagram com- mutes: [ || [ T | [[] | |u _____:w (As in Definition 1.1.1, T is the twist map.) is cocommutative if the dual dia* *gram commutes; isLbicommutative if it is both commutative and cocommutative. We say that = ii is connected if i = 0 when i < 0, and j :k -! 0 is an isomorphism. Note that if each homogeneous piece iis finite-dimensional, then the graded dual * of has the structure of a Hopf algebra. Then is commutative if and only if * is cocommutative, and so forth. Example 1.1.4. (a)The homology of a topological group G with coeffi- cients in a field k is a cocommutative Hopf algebra; it is connected if and only if G is connected. (b)For any group G, the group algebra kG is a cocommutative Hopf algebra. It is commutative if and only if G is abelian. It is graded trivially: eve* *ry element is homogeneous of degree zero. If G is finite, then the vector spa* *ce dual of kG is a commutative Hopf algebra. (c)For any Lie algebra L, its universal enveloping algebra U(L) is a cocommu- tative Hopf algebra. It is commutative if and only if L is abelian. As with kG, it is graded trivially (unless L is graded itself). (d)Similarly, if k has characteristic p, for any restricted Lie algebra L, it* *s re- stricted universal enveloping algebra V (L) is a cocommutative Hopf algebr* *a. (e)Fix a prime p. The mod p Steenrod algebra A* is a cocommutative Hopf algebra; its dual A* is a commutative Hopf algebra. Both A* and A* are connected. Starting in the next chapter, we will focus almost exclusively * *on this example. 4 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA Definition 1.1.5.An element fl of a Hopf algebra is primitive if (fl) = fl 1 + 1 fl. We write P for the vector space of all primitives of . We say th* *at fl is grouplike if (fl) = fl fl. 1.1.2. Comodules. We move on to a brief discussion of modules and comod- ules. Again, [MM65 ] is one of the standard references; [Boa ] is also quite u* *seful. Definition 1.1.6.Let be a coalgebra over a field k. A k-space M is a (left) -comodule if there is a structure map :M -! M, called the coaction map, making the following diagram commute: M ----! M: ? ? ?y ?y1 M --1--! M (In other words, defines a "coassociative" coaction.) A (left) module over a * *k- algebra is defined dually, of course. We say that M = iMi is bounded below if Mi= 0 for all sufficiently small i. Throughout, we use left comodules and left modules; from here on, we will omit the word "left." Given two -comodules M and N over a Hopf algebra , then M N is naturally a -comodule, via the structure map M N --M--N-! M N -1T1---! M N -11---! M N: This is called the "diagonal" coaction. We can also put the "left" coaction on M N, in which the structure map is M N --M-1-! M N: We rarely use this comodule structure; when we do, we denote the comodule by L M N. If we want to explicitly distinguish the diagonal coaction from the left coaction, we write M N for the tensor product with the diagonal coaction. We will use the following lemma once or twice. It is fairly standard; see [B* *oa , 5.7], for example. Lemma 1.1.7.Let M be a -comodule. Then M with the diagonal coaction L is naturally isomorphic, as a -comodule, to M with the left coaction. In particular, M is a direct sum of copies of . Proof. One can check that the following two composites are mutually inverse -comodule maps: L 1 N 1O1 N ----! N ----! N -! N; L N -1-N--! N -1-! N: |___| Lemma 1.1.8.Let be a Hopf algebra over k, and assume that each i is finite-dimensional. Let * denote the (graded) dual of . Then every -comodule 1.1. RECOLLECTIONS 5 has a natural *-module structure. If is finite-dimensional, then the categories -Comod and *-Mod are equivalent. Proof. Let M be a -comodule. Then we make it a *-module via the struc- ture map * M -1--!* M -ev1--!k M = M: If is finite-dimensional, then one chooses dual bases (fli) and (gi) for and * **. Given a *-module N with structure map ': * M -! M, we make N into a -comodule via the map N -! N; X n 7-! flj nj; j where we sum over all j so that n is "hit" by gj: we have '(gjnj) = n. Since is finite-dimensional, this is a finite sum, and hence an element of the tensor_pr* *oduct. We leave the rest of the proof to the reader. |__| 1.1.3. Homological algebra. Now we discuss a little homological "coalge- bra." [Boa ] is a good reference for this material, as well. One can also dua* *lize discussions of homological algebra for modules, as found in any number of places (such as [CE56 , Wei94, Ben91a]). Since we are working with comodules rather than modules, we work with the notions of cofree and injective comodules, which are dual to the notions of fre* *e and projective, respectively. Definition 1.1.9.Let be a k-coalgebra. A comodule M is injective if the functor Hom (-; M) is exact. A comodule M is projective if Hom (M; -) is exact. The forgetful functor U: -Comod -!k-Mod has a right adjoint, C: Hom k(UM; N) ~=Hom (M; CN): CN is called the cofree comodule on N. See [Boa ] for the following two results. Lemma 1.1.10.A comodule is injective if and only if it is a summand of a cofree comodule. Lemma 1.1.11. is cofree, and hence injective, as a -comodule. In other words, every comodule map M -! is adjoint to a vector space map M -!k, and hence is determined by the preimage of 1 2 . Injective comodules are much more important to us than projective comodules, because of the following. Example 1.1.12. (a)On one hand, if is finite-dimensional, then injective comodules are the same as projective comodules. (b)On the other hand, if = A* is the dual of the mod p Steenrod algebra, then it seems likely that there are no nonzero projective comodules. Essentiall* *y, an element in a projective comodule over A* should need to have infinitely many elements in its diagonal, so it would have to be viewed as a completed comodule, not a comodule proper. 6 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA By Lemma 1.1.7, M is injective for any comodule M; indeed, M is isomorphic to C(UM) . The map M -! M which is adjoint to the identity on UM gives us the start of an injective resolution of M. Definition 1.1.13.Given a comodule M, a sequence of injective comodules Io = (I0 -!I1 -!I2 -!: :):is an injective resolution of M if the sequence 0 -!M -!I0 -!I1 -!I2 -!: : : is exact. As is well-known, injective resolutions are unique up to chain homotopy equi* *va- lence. We are interested in injective comodules because they are useful in comp* *uting various derived functors, particularly Ext. Definition 1.1.14.Let be a Hopf algebra over k, and let M and N be -comodules. Then Exts(M; N) is the sth derived functor of Hom (M; N). To compute it, one takes an injective resolution Io of N, and defines Exts(M; N) to be the sth cohomology group of the cochain complex Hom (M; Io). Throughout this book, when we say Ext, we mean Ext in this setting: the category of comodules over a Hopf algebra. Lemma 1.1.15.Let be a connected Hopf algebra over a field k. Then the vector space Ext1(k; k) is isomorphic to P, the space of primitives of . Proof. This is dual to the statement that in the category of modules over an algebra, Ext1is in bijection with the generators of the algebra. One proves thi* *s by constructing the first few terms of a minimal resolution of k as a -comodule._We leave the details to the reader. |__| We point out that for any comodule, there is a canonical injective resolutio* *n, known as the cobar complex. We will do a few computations with it in Appendix B* *.3; we refer the reader to [Ada56 , HMS74 ] for details. 1.2.The category Stable() In this section, we define the category in which we work, and we introduce notation which we will use for the rest of this book. Let be a graded commutative Hopf algebra over a field k; we work in the category Stable() whose objects are cochain complexes of injective graded left - comodules, and whose morphisms are cochain homotopy classes of graded maps. We call objects of this category spectra, and we write the morphisms from X to Y as [X; Y ]. We will omit the word "graded" from this point on; all comodules and maps are understood to be graded. The category Stable() is a stable homotopy category in the sense of [HPS97 ]; hence one can perform many standard stable homotopy theoretic constructions in Stable(). For instance, rather than having exact sequences, one has "exact triangles," also known as "cofibrations" or "cofiber sequences." We freely use * *other language and results from [HPS97 ], often without explicit citation. Stable() is generated by the injective resolutions of the simple comodules_that is, if X is an object so that [S; X]** = 0 whenever S is an injective resolution of a simple comodule, then X is a contractible cochain complex. If the trivial comodule k is the only simple (say, if is connected, or if is the dual of the mod p group a* *lgebra 1.2. THE CATEGORY Stable() 7 of a p-group), then we say that Stable() is monogenic, at least in the graded s* *ense. In this case, the stable homotopy constructions are even more familiar. If Stable() is monogenic and the homotopy of the sphere object (defined belo* *w) is countable, then Stable() is a Brown category, so that homology functors are representable. This is the case when = A, the dual of the Steenrod algebra. (Stable() can be a Brown category even if it is not monogenic; since we are foc* *using on the Steenrod algebra in this book, though, we often assume monogenicity.) Remark 1.2.1. (a)Mahowald and Sadofsky studied the category Stable() in their paper [MS95 ], with = A. (b)If is finite-dimensional, then one could just as well work with the cat- egory of cochain complexes of injective *-modules, because in the finite- dimensional case, the categories of -comodules and *-modules are equiv- alent, with injective comodules corresponding to injective modules. When * is not finite-dimensional, in particular when * = A* is the Steenrod al- gebra, there are technical problems with the category of cochain complexes of injective A*-modules. For example, there are no maps from F2to A*, so the "homotopy" of the (injective) module A* would be zero; therefore in the module setting, we would not have the implication ss*X = 0 ) X = 0. (c)We also note that, regardless of the dimension of , the category Stable() * *is rather different from the derived category of -comodules, because homology isomorphisms are not necessarily invertible in Stable(). For instance, if = F2[x]=(x2) with x primitive, then the (periodic) cochain complex : :-:-! --! --! --! --!: :;: x 7! 1; x 7! 1; has no homology, and hence is zero in the derived category. On the other hand, it is non-contractible in Stable(); if we write Ext**(F2; F2) = F2[v* *], then this complex is a ring spectrum with homotopy groups (as defined below) equal to F2[v; v-1]. The category Stable() has arbitrary coproducts; we use the symbol _ to denote the coproduct. For objects X and Y of Stable(), we write X ^ Y for X k Y , and we call this the smash product of X and Y . This operation is commutative, associative, and unital: if S is an injective resolution of the trivial comodule k, then S i* *s the unit of the smash product. We call S the sphere spectrum. We grade morphisms as follows: first of all, we let [X; Y ]0;0= [X; Y ]. We write S = (I0 -!I1 -!I2 -!: :):; and we let Si;jbe the cochain complex which is s-jIn+iin homological degree n (here s denotes the "internal" suspension functor s: -Comod -!-Comod). For integers i and j we define the (i; j)-suspension functor i;jby i;j:Stable() -!Stable(); X 7-! Si;j^ X: L Then we let [X; Y ]i;j= [i;jX; Y ]0;0, and we write [X; Y ]**for i;j[X; Y ]i;* *j. Remark 1.2.2.The grading is the usual Ext grading: if X and Y are injective resolutions of comodules M and N, respectively, then [X; Y ]i;j= Exti;j(M; N). 8 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA Hence if is concentrated in degree 0 (e.g., if = (kG)*), then one may as well work with -comodules concentrated in degree 0, in which case [-; -]ij= 0 if j 6= 0, and [-; -]i0= Exti(- ; -). We will follow the (somewhat odd) tradition * *in homotopy theory of drawing pictures of Ext (and hence of [-; -]) using the Adams spectral sequence grading: Exts;tis drawn with s on the vertical axis and t - s* * on the horizontal axis. Unfortunately, because of the form of long exact sequences in Ext, cofiber sequences look like this: : :-:!1;0Z -!X -!Y -! Z -!-1;0X -!: ::: So one needs to take a little care when translating proofs from ordinary homo- topy theory to this setting. Given a spectrum X, we define the homology functor associated to X, Xij, by Xij:Stable() -!Ab; Y 7-! [S; X ^ Y ]i;j; and we define the cohomology functor associated to X, Xij, by Xij:Stable()op-! Ab; Y 7-! [Y; X]-i;-j: When X = S, we have a special notation for Xij: we define the (i; j)-homotopy group of Y to be ssijY = SijY = [S; Y ]ij. Note that if X -! Y -! Z is a cofiber sequence, then we have : :-:!ssi-1;jZ -!ssi;jX -!ssi;jY -! ssi;jZ -!ssi+1;jX -!: : : (andLsimliarly for other homology functors). As with morphisms, we write ss**(-) for i;jssi;j(-) (and similarly for other homology and cohomologyLfunctors). A* *lso, given a spectrum X, we write Xi;jfor ssi;jX = Xi;jS, and X**for i;jXi;j. We say that an object R in Stable() is a ring spectrum if there is a multipl* *ica- tion map : R ^R -!R and a unit map S -!R, making the appropriate diagrams commute. We often abuse notation and let S0 = S0;0= S. One of our main goals is to get as much information as possible about ss**S0 = Ext**(k; k). 1.3. The functor H We assume that is a graded commutative Hopf algebra over a field k, and we work in the category Stable(). The quotient coalgebras and Hopf algebras of carry useful information; in this section we construct a spectrum HB in Stable() for each quotient coalgebra B of , and we study the properties of the functor H. Recall that if B is a quotient coalgebra of and if M is a B-comodule, then the cotensor product 2BM is defined to be the equalizer of the two maps M -1--M--! B M; M ---1M--! B M: Here is the right B-comodule structure map on , and M is the left B-comodule structure map on M. (The tensor products are over k.) 1.3. THE FUNCTOR H 9 Definition 1.3.1.We define a covariant functor H from quotient coalgebras of to spectra by defining HB to be an injective resolution of 2Bk. For example, Hk = , so that Hk**(X) is the homology of the cochain com- plex X; also, H = S0. H provides a useful source of (co)homology functors on Stable(). The general philosophy is that if one has a quotient B of , rather th* *an studying B by working in Stable(B), one studies B by looking at HB in the cate- gory Stable(). This is borne out by Corollary 1.3.5, as well as the other resul* *ts in this section. Fix a quotient coalgebra B of . Given an -comodule M, we let M#B denote its restriction to B; this is the B-comodule with structure map M -! M -! B M. Recall from [MM65 ] and [Rad77 ] that if B happens to be a quotient Hopf algebra of , then #B is injective as a right (and a left) B-comodule. Proposition 1.3.2.For any quotient coalgebra B of over which #B is in- jective as a right B-comodule, we have HBij~=ExtijB(k; k). Furthermore, we have the following. (a)If B is a quotient Hopf algebra of , then HB is a commutative ring spec- trum. (b)If B is a quotient coalgebra of B over which #B is injective as a right comodule, then HB has many of the properties of a ring spectrum: (i)HB**is a k-algebra, and for any spectrum Y , HB**Y is a right module over HB**. (ii)There is a "unit map" S0 -! HB which induces an algebra map ss**S0 -!HB**. (iii)More generally, if B i C are quotient coalgebras of over which is injective, then the induced map HB**-! HC**is an algebra map. Proof. That HB**~=Ext**B(k; k) follows from Lemma 1.3.4 below. Part (a) is clear_since is commutative as an algebra, then so is B, and the commutative product on B induces one on HB. The quotient map i B induces the unit map S0 = H -! HB, and hence a Hurewicz map ss**(X) -! HB**(X). This Hurewicz map is the same as the restriction map res;B: Ext**(k; k) -! Ext**B(k; k). For part (b), the unit map is induced from the quotient i B, as in (a). The induced map in (iii) is just the restriction map. The rest of (b) follows_f* *rom Lemma 1.3.4 and Corollary 1.3.5. |__| Example 1.3.3.Note that if B is a quotient coalgebra of , then HB need not be a ring spectrum. For example, suppose that k has characteristic p, and l* *et = k[x] with x primitive. Then B = k[xp] is a quotient coalgebra of , and HB is an injective resolution of M = k[x]=(xp). A multiplication on HB would induce one on M (by taking homology), and it is easy to see that M is not a -comodule algebra: the comodule structure on M is given by x 7! 1 x + x 1 2 M, so if this coaction were multiplicative, we would have 0 = xp 7-! xp 1 6= 0: Lemma 1.3.4.Suppose that B is a quotient coalgebra of so that #B is in- jective as a right B-comodule. (a)Given a -comodule M and a B-comodule N, we have an isomorphism Ext**B(M#B ; N) ~=Ext**(M; 2BN ): 10 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA (b)If N is a -comodule, then we have an isomorphism of -comodules 2BN ~=( 2Bk) N: (c)Hence if M and N are both -comodules, then we have Ext**B(M#B ; N#B) ~=Ext**(M; ( 2Bk) N ): Proof. Part (a) is (the coalgebra version of) Shapiro's lemma. See [Wei94 , Lemma 6.3.2], [Ben91a , Corollary 2.8.4], or any other good reference on homolo* *g- ical algebra, for a statement in terms of modules and algebras. We leave it to * *the reader to dualize to our version; see [Rav86 , A1.3.13] for a related result. Part (b), the "shearing isomorphism," seems to be well-known, but it may be difficult to locate a proof in the published literature (see [HSb ] and [Boa ],* * for instance). Let Ntrdenote N with the trivial coaction of , so that Ntris the L tensor product N with the left coaction (as compared with N = N, the tensor product with the diagonal coaction). The desired isomorphism is induced * *by one between Ntrand N, because 2BN is a subcomodule of the former, and ( 2Bk) N a subcomodule of the latter. Hence Lemma 1.1.7 completes the proof of (b). __ Part (c) follows from parts (a) and (b). |__| Given a quotient coalgebra B of and given objects X and Y of Stable(), we let [X; Y ]B denote the set of cochain homotopy classes of maps from X to Y , viewed as cochain complexes of left B-comodules. Corollary 1.3.5.Suppose that B is a quotient coalgebra of so that #B is injective as a right B-comodule. Given objects X and Y of Stable(), we have [X; HB ^ Y ]**~=[X; Y ]B**. In particular, HB**is an algebra and HB**Y is a rig* *ht module over it. Corollary 1.3.6.The spectrum Hk is a field spectrum: it is a ring spectrum, and any Hk-module spectrum is a wedge of suspensions of Hk. In particular, for any X, Hk ^ X is a wedge of suspensions of Hk. Proof. By Proposition 1.3.2(b), Hk = is a ring spectrum. By Lemma 1.1.7, for any -comodule M, M is a direct sum of copies of ; one can check (by [Boa , 5.7], for instance) that given a comodule map M -! N, the induced map M -! N sends each summand of either isomorphically to a summand, or to zero. So if X is any cochain complex of -comodules, then Hk ^ X = X splits into a direct sum of cochain complexes of the forms 0 -! =-! -!0 and 0 -! -!0: |___| 1.3.1. Remarks on Hopf algebra extensions. Many results about group cohomology (and indeed about Hopf algebra cohomology in general) are proved using the spectral sequence associated to an extension. If one has a group exte* *nsion in which the quotient is cyclic of prime order, the associated spectral sequenc* *e is particularly tractable, and hence quite useful. In this subsection we remind t* *he reader of standard notation related to Hopf algebra extensions, and then we foc* *us 1.3. THE FUNCTOR H 11 on extensions with small kernel. In Section 1.5 we discuss the spectral sequence associated to a general Hopf algebra extension. Definition 1.3.7. (a)Suppose that C ,! is an inclusion of augmented algebras over a field k, and let IC be the augmentation ideal of C_the ker* *nel of C -!k. We say that C is normal in if the left ideal of generated by C (i.e., . IC) is equal to the right ideal generated by C (i.e., IC . ). * *If C is normal in , then we let ==C = C k = k C . In this case, C -! -!==C is an extension of augmented k-algebras. (b)Dually, suppose that -!B is a surjective map of coaugmented coalgebras over k, and let JB denote the coaugmentation coideal of B_the cokernel of k -!B. We say that B is a conormal quotient of if 2Bk = k 2B. If B is a conormal quotient of , then 2Bk -! -!B is an extension of coaugmented k-coalgebras. As far as this definition goes, for us will usually be a Hopf algebra with commutative multiplication, so that every subalgebra of will be normal. We will be more interested in quotients of ; if -!B is a surjective map of commutative Hopf algebras over k, then 2Bk is the algebra kernel. So B is conormal if the algebra kernel is also a subcoalgebra of . Remark 1.3.8.If is a commutative Hopf algebra and B is a quotient coalge- bra of over which #B is injective as a right comodule, then Lemma 1.3.4 tells * *us that Ext**B(k; k) ~=Ext**(k; 2Bk). If, in addition, B is conormal, then there * *is a right coaction of on 2Bk, which induces a left coaction of on Ext**B(k; k). * *The coaction of B is clearly trivial, so we get a left coaction of 2Bk on Ext**B(k* *; k); indeed, we get a left coaction of 2Bk on Ext**B(k; M) for any left B-comodule * *M. For the remainder of this section, we assume that the ground field k has cha* *r- acteristic p > 0. Notation 1.3.9.Given a homogeneous element x in a graded vector space, let |x| denote its degree. Let E[x] denote the Hopf algebra k[x]=(x2) with x primit* *ive, where |x| is odd if p is odd. Let D[x] = k[x]=(xp) with x primitive, where |x| is even if p is odd. Recall from Lemma 1.1.15 that if B is a Hopf algebra, then Ext1;*B(k; k) = HB1;*is isomorphic to the vector space of primitives of B. If y* * 2 B is primitive, we let [y] denote the associated element of HB1;*. Recall further* * that if p is odd, then there is a Steenrod operation fifP0: Exts;tB(k; k) -!Exts+1;ptB(k; k): (See [May70 , Wil81], as well as Appendix B.1.) Here is our analysis of Hopf algebra extensions with small kernel. Lemma 1.3.10.Fix a graded commutative Hopf algebra over a field k of char- acteristic p > 0. (a)Suppose that there is a Hopf algebra extension of the form E[x] -!B -!C; 12 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA where B is a quotient Hopf algebra of . Then x is primitive in B, so there is a nonzero element h = [x] in Ext1;|x|B(k; k) = HB1;|x|. We also let h: 1;|x|HB -!HB denote the corresponding self-map of HB_i.e., the composite 1;|x|HB = S1;|x|^ HB h^1--!HB ^ HB -!HB; where is the multiplication map. Then there is a cofiber sequence 1;|x|HB h-!HB -!HC -!0;|x|HB: (b)Suppose that p is odd and there is a Hopf algebra extension of the form D[x] -!B -!C; where B is a quotient Hopf algebra of . Then x is primitive in B; we let b = fifP0[x] in Ext2;p|x|B(k; k) = HB2;p|x|. We also let b: 2;p|x|HB -!HB denote the corresponding self-map of HB. Then there is a cofiber sequence 2;p|x|HB -b!HB -!gHC -!1;p|x|HB; where gHCis defined by a cofibration 1;|x|HC -!gHC -!HC -!0;|x|HC: The element b = fifP0[x] in (b) is also the p-fold Massey product of [x] with itself, as mentioned in [May70 , 11.11]. Proof. Part (a): It is clear that x is primitive, so we only need to discuss* * the putative cofibration. There is a short exact sequence of E[x]-comodules 0 -!k -!E[x] -!|x|k -!0: We apply the (exact) functor 2B- to this, noting that E[x] = B 2Ck: 0 -! 2Bk -! 2Ck -!|x| 2Bk -!0: Taking injective resolutions then gives the desired cofibration. Part (b): Note that there are two short exact sequences of D[x]-comodules: 0 -!k -!D[x] -!|x|k[x]=(xp-1) -!0; 0 -!k[x]=(xp-1) -!D[x] -!(p-1)|x|k -!0: We apply the functor 2B- to these and take injective resolutions; writing gHB for the injective resolution of 2B(k[x]=(xp-1)), we have the following cofiber* * se- quences: 1;|x|gHBj-!HB -!HC -!0;|x|gHB; 0 1;(p-1)|x|HB j-!gHB-! HC -!0;(p-1)|x|HB: It is standard (e.g., [Ben91b , pp. 137-8]) that the map b is the composite j O* * j0, and the 3 x 3 lemma (or the octahedral axiom_see [HPS97 , A.1.1-A.1.2]) allows * * __ us to identify the cofiber of j O j0in terms of the cofibers of j and j0. * * |__| 1.4. SOME CLASSICAL HOMOTOPY THEORY 13 1.4.Some classical homotopy theory We assume that is a graded commutative Hopf algebra over a field k, and we work in the category Stable(). For this section, we assume that Stable() is monogenic (i.e., the trivial comodule k and its suspensions are the only simple comodules). We also assume that is non-negatively graded: n = 0 if n < 0. Because ssijS0 = Exti;j(k; k), then ss**S0 is concentrated in the first quad* *rant. More precisely, ssijS0 = 0 if j < 0 and (unless i = j = 0) if i 0; furthermore, ss00S0 = k. Since ss**S0 is "connected" in this sense, we can construct cellul* *ar towers and hence Postnikov towers, as in the usual stable homotopy category (and indeed in any connective stable homotopy category_see [HPS97 , Section 7]). We also have a Hurewicz theorem. Since we are working in a bigraded setting rather than the singly graded setting of [HPS97 , Section 7], we state the relevant re* *sults; we leave the proofs as an exercise for the reader. (See also the appropriate pa* *rts of [Mar83 ].) Definition 1.4.1.Given a spectrum X, we say that : :-:!Xn-1 -!Xn -!Xn+1 -!: : : is a cellular tower for X if (i)wcolimXn = X, (ii)The fiber of Xn -!Xn+1 is a coproduct of spheres Si;jwith i + j = n, (iii)lim-Hk**Xn = 0. Because of the connectivity properties of ssijS0, we have the following theo* *rem, guaranteeing existence of cellular and Postnikov towers. Theorem 1.4.2. (a)Any spectrum X has a cellular tower. (b)For any spectrum X and integer n, there is a cofiber sequence X[n; 1] -!X -!X[-1; n - 1]; so that (i)ssijX[n; 1] = 0 if i + j < n, (ii)ssijX[-1; n - 1] = 0 if i + j n, (c)If for some integer n we have spectra X and Y with X = X[n; 1] and Y = Y [-1; n - 1], then [X; Y ]**= 0. (d)Hence for any spectrum X, we can construct a Postnikov tower: : :-:---! X[-1; r]----! X[-1; r - 1]----! X[-1; r - 2]----!: : : x? x x ? ?? ?? X[r] X[r - 1] X[r - 2] (Here X[r] is a coproduct of factors of the form i;jHk, where i + j = r.) The sequential colimit of X[-1; r] is 0, and the sequential limit of X[-1;* * r] is X. (e)Dually, we can construct a diagram : :-:---! X[r; 1]----! X[r - 1; 1]----!X[r - 2; 1]----!: : : ?? ? ? y ?y ?y X[r] X[r - 1] X[r - 2] 14 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA The sequential colimit of the "connective covers" X[r; 1] is X, and the sequential limit is 0. Parts (d) and (e) are easy in our context: if k is the only simple comodule,* * then every injective is a direct sum of copies of = Hk. So to construct X[r; 1], for example, one just truncates the cochain complex X at bidegrees (s; t) with s+t * *< r. Note that if is a connected Hopf algebra (Definition 1.1.3), then ssijS0 = 0 if j < i; since the Steenrod algebra is connected, we use this pattern for our definition of connectivity. At the other extreme, if is concentrated in degree* * zero, then ssstS0 = 0 if t 6= 0; this leads to a weaker notion of connectivity. Definition 1.4.3.Given a spectrum X, if there exist numbers i0 and j0 so that ssijX = 0 when i < i0 or j - i < j0, then we say that X is (i0; j0)-connec* *tive. We say that X is connective if X is (i0; j0)-connective for some unspecified i0* * and j0. If for some i0 and j0, we have ssijX = 0 when i < i0 or j < j0, we say that* * X is weakly (i0; j0)-connective. Here is the second main theorem of this section, a bigraded version of the Hurewicz theorem. Theorem 1.4.4.If X is (i0; j0)-connective, then the Hurewicz map ssijX -! HkijX is an isomorphism when i < i0 or j - i j0. Similarly, if X is weakly (i0; j0)-connective, then ssijX -!HkijX is an isomorphism when i < i0 or j j0. Remark 1.4.5.We point out that one can use the bigrading to generalize these results somewhat; for example, one can discuss cellular towers with "slope u_v"_ towers as above, but with the fiber of Xn -!Xn+1 equal to a coproduct of spheres Si;jwith vi + uj = n. Using the form of connectivity of the sphere spectrum, one can construct cellular towers of any given nonpositive slope. Similar remarks h* *old for Postnikov towers. Similarly, one can define "connectivity with slope u_v" for nonpositive u_v_* *i.e., ss**X = 0 below a line of slope u_v_and then show that the Hurewicz map is an isomorphism at and below the line of connectivity. Furthermore, if happens to be connected, then ssijS0 = 0 when j < i and when i < 0. In this case, one can construct cellular towers and prove Hurewicz theorems for any slope m 0 or m > 1. One could also work with connectivity determined by arbitrary non-increasing curves rather than lines of nonpositive slope and get the same sorts of results* *. We have no need to work with anything approaching this level of generality. If is connected and X is an injective resolution of a bounded below comod- ule M, then X is (0; 0)-connective, but it also satisfies a stronger property. * *This property occurs several times in this work, so we make it into a definition. Definition 1.4.6.We say that a spectrum X is comodule-like, or CL, if X satisfies the following conditions: (i)There exists an integer i0 such that ssi*X = 0 if i < i0, (ii)There exists an integer j0 such that ssijX = 0 if j - i < j0, (iii)There exists an integer i1 so that (Hk)i*X = 0 for i > i1. (When X is an injective resolution of M, we may take i0 = 0 = i1, and j0 to be the degree of the bottom class of M.) 1.5. THE ADAMS SPECTRAL SEQUENCE 15 Note that X is a CL-spectrum if and only if X has a cellular tower built of spheres Si;jwith i0 i i1 and j - i j0. We will need the following lemmas later. First we need to recall a few defin* *itions from [HPS97 , 1.4.3, 2.1.1]. Definition 1.4.7. (a)A full subcategory D of Stable() is localizing if it is "closed under cofibrations and coproducts": if X -!Y -! Z is a cofibration and two of X, Y , andLZ are in D, then so is the third; if {Xff} is a set * *of objects in D, then Xffis in D. Given an object Y , we let loc(Y ) denote the localizing subcategory generated by Y , i.e., the intersection of all * *of the localizing subcategories containing Y . (b)Similarly, a full subcategory D of Stable() is thick if it is closed under cofibrations and retracts (if Y is in D and there are maps X -!Y -! X so that the composite is an isomorphism, then X is in D); and thick(Y ) denot* *es the thick subcategory generated by Y . (c)A property P of spectra is generic if the full subcategory of spectra sati* *sfying P is thick. (d)An object X of Stable() is finite if and only if it is small (in the categ* *orical sense), if and only if it is in thick(S0). If is connected, then an object X is finite if and only if X is connective * *and has dimkHk**X < 1. So if is connected and B is a quotient Hopf algebra of , then an object X of Stable() is finite if and only if its restriction X#B is fi* *nite in Stable(B). Lemma 1.4.8.Suppose that X is a spectrum and that there is a line of nonpos- itive slope above which the homotopy of X is zero (i.e., for some u; v with u_v* * 0, there is an n so that if vi + ij n, then ssijX = 0). Then X is in the localizi* *ng subcategory generated by = Hk. Proof. The Postnikov tower of slope u_vfor such an X displays X as being a * * __ colimit of objects of loc(); hence X is itself an object of loc(). * *|__| Lemma 1.4.9.If D is a localizing subcategory of Stable() which contains a nonzero finite spectrum, then loc() D. Proof. It suffices to show that 2 obD if D is as given. Let Y be a nonzero finite object of D; then Y ^ is nonzero (by the Hurewicz theorem 1.4.4_remember that = Hk) and is contained in obD. On the other hand, Corollary 1.3.6 tells us that Y ^ is a direct sum of suspensions of , so by the Eilenberg swindle_[HPS97* * , 1.4.9], 2 obD. |__| We will see in Corollary 4.5.7 that as long as is infinite-dimensional, the containment loc() D is strict (i.e., loc() contains no nonzero finite spectrum* *). 1.5.The Adams spectral sequence As in the rest of this chapter, we assume that is a graded commutative Hopf algebra over a field k. We discuss the generalized Adams spectral sequence associated to the homology theory HB**in this section, for B a conormal quotient of (see Definitions 1.3.1 and 1.3.7). In particular, we note that it is the sa* *me as the spectral sequence associated to a Hopf algebra extensions, and we derive a * *few consequences. 16 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA For the general approach to the Adams spectral sequence, see [Ada74 ]. We also recall a construction of the spectral sequence in Appendix A.2. Theorem 1.5.1.Suppose that B is a conormal quotient Hopf algebra of and fix a spectrum X. Then the Adams spectral sequence based on the homology theory HB**has E2-term Es;t;u2= Exts;t;uHB**HB(HB**; HB**X); with differentials dr: Es;t;ur-!Es+r;t+r-1;u-r+1r; and abuts to sss+u;t+uX. Given a -comodule M, the change-of-rings spectral sequence associated to the extension 2Bk -! -!B has E2-term 8Ep;q;v2= Extp;v q;* (1.5.2) 2Bk (k; ExtB(k; M)) and converges to Extp+q;v(k; M): (The action of 2Bk on Ext**B(k; M) was discussed in Remark 1.3.8.) The differ- entials are indexed as follows: dr: 8Ep;q;vr-!8Ep+r;q-r+1;vr: (See [Sin73, II, x5], for example; alternatively, one can dualize the construct* *ion of the Lyndon-Hochschild-Serre spectral sequence for the computation of group cohomology. See [Ben91b , 3.5] for a construction of this as the spectral seque* *nce associated to a double complex, for instance.) Proposition 1.5.3.Suppose that B is a conormal quotient of , and suppose that X is an injective resolution of a -comodule M. Then the HB-based Adams spectral sequence abutting to ss**X is isomorphic, up to a regrading, to the ch* *ange- of-rings spectral sequence associated to the extension 2Bk -! -!B; abutting to Ext**(k; M). The regrading is as follows: for all r 2, the Es;t;ur* *-term of the Adams spectral sequence is isomorphic to the 8Es;u;t+ur-term of the chan* *ge- of-rings spectral sequence. An example of the regrading is the isomorphism of E2-terms Extp;v2Bk(k; Extq;*B(k;)M)~=Extp;v-q;qHB**HB(HB**; HB**X): Proof. This is an exercise in homological algebra. The key observation is th* *at since B is conormal, then 2Bk is a trivial B-comodule. Hence HB**HB = Ext**B(k; 2Bk) ~=( 2Bk) Ext**B(k; k) = 2Bk HB**: If X is an injective resolution of a -comodule M, then combining X with an Adams tower for X yields a double complex; using the above isomorphism, one can easily show that the resulting spectral sequence is the change-of-rings spectral seque* *nce. 1.5. THE ADAMS SPECTRAL SEQUENCE 17 The shift in gradings comes from a more precise statement of the above iso- morphism: M `;j Ext`;mB(k; 2Bk) ~= ( 2Bk)i ExtB(k; k): i+j=m |___| Hence the HB-based Adams spectral sequence has some nice properties. For example, we have a convergence result: if X and M are as in the proposition, th* *en the Adams spectral sequence converges to ss**X = Ext**(k; M). (We also show in Proposition A.2.5 that the spectral sequence converges to ss**X whenever is a connected Hopf algebra and X is connective_i.e., every connective spectrum is "HB-complete".) Also, if X = S0 (or more generally if X is an injective resolution of a comm* *uta- tive -comodule algebra), then one has Steenrod operations acting on this spectr* *al sequence, as described in [Sin73] (see also [Saw82 ]). We need the following re* *sult in Chapter 3. Proposition 1.5.4.Suppose that is a Hopf algebra over the field Fp, and suppose that B is a conormal quotient of . Consider the HB-based Adams spectral sequence converging to ss**S0. Givennyn2 E0;t;u2, with t and u even if p is odd* *, then for each n, ypn survives to E0;ppt;pnu+1. (Note that the result is the same whether one is using the Adams grading (Theorem 1.5.1) or the change-of-rings grading (equation (1.5.2)): the elements* * in 8E0;q;v2in the change-of-rings grading correspond to elements in E0;v-q;q2= E0;* *t;u2 in the Adams grading; those with q and v even correspond to those with t and u even.) Proof. This follows from properties of Steenrod operations on this spectral sequence, as discussed in [Sin73] and [Saw82 ]. Suppose we have a spectral se- quence Es;trwhich is a spectral sequence of algebras over the Steenrod algebra.* * Fix z 2 Es;tr, and fix an integer k. If p = 2, then [Sin73, 1.4] tells us to which * *term of the spectral sequence Sqkz survives (the result depends on r, s, t, and k). * *For instance, if z 2 E0;tr, then Sqt(z) = z2 survives to E0;2t2r-1. [Saw82_,_2.5] * *is the corresponding result at odd primes. |__| By the way, Singer's results [Sin73] are stated in the case of an extension * *of commutative Hopf algebras B -! -!C where C is also cocommutative (actually, he works in the dual situation). This cocommutativity condition is not, in fact, necessary, as forthcoming work of Si* *nger shows [Sin]. We also need the theorem of Hopkins and Smith [HSa ] that the presence of a vanishing plane in the Adams spectral sequence is a generic property. Suppose that E is a spectrum satisfying the following conditions (cf. [Ada74 , III.15] * *and [Rav86 , 2.2.5]): (a)E is a commutative associative ring spectrum. (b)E**E is flat over E**. 18 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA (c)E is (0; 0)-connective (Definition 1.4.3), and the unit map j :S0 -!E in- duces an isomorphism on ss0;0. (d)HkijE is a finite-dimensional k-vector space for each i and j. Theorem 1.5.5 (Hopkins-Smith).Suppose that is a connected Hopf algebra. Consider the Adams spectral sequence Es;t;urbased on a ring spectrum E satisfyi* *ng the above conditions. Fix numbers m 0 and n. The full subcategory of Stable() consisting of all spectra X that satisfy the property "there exist numbers r an* *d b so that Es;t;ur(X) = 0 when s m(s + u) + n(t + u) + b" is thick. Since we are not aware of any published proof of this, we give a proof in Appendix A.2. 1.6.Bousfield classes and Brown-Comenetz duality Again, we suppose that is a graded commutative Hopf algebra over a field k. We assume that Stable() is monogenic. In this section we collect some useful results about Bousfield classes, Brown-Comenetz duality, and their interaction. We remind the reader that the Bousfield class of an object X in an arbit* *rary stable homotopy category is the collection of X-acyclic objects_all objects Z w* *ith X ^ Z = 0. We order Bousfield classes by reverse inclusion, so means that X ^ Z = 0 ) Y ^ Z = 0. X and Y are Bousfield equivalent if = . We define the operation _ on Bousfield classes by _ = ; one can s* *how that this is the least-upper bound of and . A theorem of Ohkawa [Ohk89 ] says that there is a set of Bousfield classes; hence there is also a greatest-l* *ower bound operation (take the least-upper bound of all the classes less than or equ* *al to both and ). For certain well-behaved classes of spectra, the greatest-l* *ower bound is given by ^, but that is not true in general. These concepts were introduced (in the ordinary stable homotopy category) by Bousfield in [Bou79a ] and [Bou79b ]; Ravenel proved a number of fundamen- tal results about them in [Rav84 ]. See [HPS97 , Section 3.6] for a discussion * *of Bousfield classes in a general stable homotopy category. For any spectrum X, we define its Brown-Comenetz dual IX to be the spectrum that represents the cohomology functor Y 7-! Hom**k(ss**(X ^ Y ); k): Hence ss**IX = Hom **k(ss**X; k). (In a general stable homotopy category, one should take Hom over R = ss0(S0) into the injective hull of R, essentially. See [BC76 ] for the analogue in the ordinary stable homotopy category.) We say that the homotopy of a spectrum X is of finite type if ssijX is finit* *e- dimensional over k for each bidegree (i; j). Proposition 1.6.1.[Rav84 ] Let X and Y be spectra. (a)Given a map f :X -! X with cofiber X=f and telescope f-1X, we have _ = and ^ = 0. (b)If X is weakly connective (Definition 1.4.3), then IX is in the localizing subcategory generated by Hk = . (c)Suppose that X has finite type homotopy. If [Y; X]**= 0, then ss**(Y ^IX) = 0. (d)If X is a ring spectrum and Y is an X-module spectrum, then we have = . 1.7. FURTHER DISCUSSION 19 (e)If X is a ring spectrum, then IX is an X-module spectrum; hence, . (f)Suppose that X is a noncontractible ring spectrum with finite type homoto* *py, and Y is an X-module spectrum. If [Y; X]**= 0, then < . See [Rav84 ] for the proofs. (See also Lemma 1.4.8 for (b).) Combining part (a) of this with Lemma 1.3.10 gives us a corollary. Corollary 1.6.2. (a)Suppose that there is a Hopf algebra extension of the form E[x] -!B -!C; giving a cofiber sequence 1;|x|HB h-!HB -!HC -!0;|x|HB: Then = _ , and HC ^ h-1HB = 0. (b)Suppose that there is a Hopf algebra extension of the form D[x] -!B -!C; with |x| > 0, giving cofiber sequences 2;p|x|HB -b!HB -!gHC -!1;p|x|HB; 1;0HC -!1;|x|HC -!gHC -!HC: Then = _ , and HC ^ b-1HB = 0. Proof. Part (a) is immediate; part (b) follows once we show that = . By the second cofibration above, it suffices to show that f-1HC is con-_ tractible. For degree reasons, though, ss**f-1HC = 0. |__| 1.7.Further discussion In this section, we note that we can apply the technology of this chapter to group algebras, and we give one or two examples. Let k be a field of characteristic p > 0, and let G be a finite group. Then * *kG is a finite-dimensional cocommutative Hopf algebra, so we can use stable homotopy theory to study Stable(kG*). By Remark 1.2.1, we could just as well work with t* *he category of cochain complexes of injective kG-modules; in either case, morphisms in the category relate to group cohomology, as mentioned in Remark 1.2.2. Since kG is concentrated in degree 0, then morphisms are only singly graded. As a result, most of the results in this chapter apply to Stable(kG*). If G is p-group, then the trivial module k is the only simple; so all of the results* * of the chapter apply to Stable(kG*) for G a p-group. To illustrate, Lemma 1.3.10 translates to the following, when p = 2. Corollary 1.7.1.Let k be a field of characteristic 2, and let G be a finite group. Any group extension 1 -!B -!G -!Z=2 -!1 gives rise to a cofibration in Stable(kG*): 1HG z-!HG -!HB; where z 2 H1(G; k) is the inflation of the polynomial generator in H1(Z=2; k). 20 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA As mentioned in Subsection 1.3.1, when one has a group extension like 1 -!B -!G -!Z=2 -!1; one can often replace arguments involving the associated spectral sequence with simpler arguments involving the cofibration in the corollary. It is an amusing * *exer- cise to prove Chouinard's theorem [Cho76 ] this way, for instance. We should also discuss the Bousfield lattice in Stable(kG*). In any stable homotopy category, the partially ordered set of Bousfield classes, together with the meet and join operations, contains important structural information about t* *he category. It turns out that when G is a finite p-group, there is a complete des* *cription of the Bousfield lattice, essentially due to Benson, Carlson, and Rickard [BCR9* *6 , BCR ]. Given a p-group G and an algebraically closed field k of characteristic p, t* *hey define VG(k) to be the maximal ideal spectrum of the (graded) commutative Noe- therian ring H*(G; k). Then for each closed homogeneous irreducible subvariety V of VG(k), they define a kG-module (V ) satisfying various properties, as out- lined below. (Throughout, they work in the category StMod(kG), so that (V ) is only well-defined up to projective summands. One could just as easily work in Stable(kG*), in which case (V ) is a cochain complex, well-defined up to chain- homotopy equivalence. Naturally, we choose the latter course.) Theorem 1.7.2 ([BCR96 ]).The objects (V ) satisfy the following: (a)There is a Bousfield class decomposition of : _ = <(V )> V VG(k) (where the wedge is taken over all closed homogeneous irreducible V ). (b)If V 6= W, then (V ) ^ (W) = 0. (c)For any V VG(k) and objects X and Y , if (V ) ^ X ^ Y = 0, then either (V ) ^ X = 0 or (V ) ^ Y = 0. By [HPS97 , 5.2.3], this implies the following. Corollary 1.7.3 ([BCR ]).When G is a p-group, the thick subcategories of finitely generated kG-modules are in one-to-one correspondence with collections* * of closed homogeneous subvarieties of VG(k) which are closed under specialization. Their result also has the following corollary. Corollary 1.7.4.Each <(V )> is a minimal nonzero Bousfield class. Hence the Bousfield lattice is a Boolean algebra on the classes <(V )>. Indeed, for e* *very X, _ = <(V )>: V :(V )^X6=0 W Proof. To show that <(V )> is minimal, we let E = W6=V(W). Then = <(V )> _ , and (V ) ^ E = 0_in other words, (V ) is complemented. If X is any object with < <(V )>, then there is a spectrum Z with X ^ Z = 0 but (V ) ^ Z 6= 0. Hence (V ) ^ X ^ Z = 0; by Theorem 1.7.2(c), this implies that (V ) ^ X = 0. Since = <(V )> _ , then we have = . On the other hand, we have (V ) ^ E = 0 and <(V )> > , so X ^ E = 0. Hence = = <0>, so X = 0. 1.7. FURTHER DISCUSSION 21 Hence for any spectrum Y , if (V ) ^ Y 6= 0, then <(V ) ^ Y > = <(V )>. Combined with Theorem 1.7.2(a), this gives the desired description of_the Bousf* *ield lattice. |__| These results describe much of the global structure of the category Stable(k* *G*) (and also of StMod(kG)), but they leave open the question of classifying all lo* *cal- izing subcategories (Definition 1.4.7). We conjecture that they are in one-to-o* *ne correspondence with arbitrary collections of closed homogeneous subvarieties of VG(k). This sort of general structure is discussed in [HPS97 , Chapter 6]. One can construct the objects (V ) in Stable() for any whose cohomology ring is Noetherian_see [HPS97 , 6.0.8, 6.1.4] (this includes all finite-dimensi* *onal commutative Hopf algebras, by work of Friedlander and Suslin [FS97 ]). The ana- logues of parts (a) and (b) of Theorem 1.7.2 hold, but it is not clear whether * *part (c) does. It is natural to conjecture that a similar description of the Bousfie* *ld lat- tice is valid, as long as the Hopf algebra is suitably well-behaved. For insta* *nce, if every quasi-elementary quotient of is elementary (these terms are defined in Section 2.1), then one might expect this. In such a situation, one could try to imitate the work in [BCR96 ]; if the quasi-elementaries do not coincide with t* *he elementaries, one still might be able to imitate Benson et al., if one had a go* *od enough understanding of the quasi-elementary quotients of . (In particular, if is a finite-dimensional quotient of the dual of the mod 2 Steenrod algebra, its quasi-elementary quotients are all elementary; if is a finite-dimensional quotient of the dual of the odd primary Steenrod algebra, the quasi-elementary quotients are more complicated. In both cases, though, we would conjecture that the analogues of 1.7.2-1.7.4 hold.) 22 1. STABLE HOMOTOPY OVER A HOPF ALGEBRA CHAPTER 2 Basic properties of the Steenrod algebra The results and constructions in Chapter 1 hold as long as is a graded comm* *u- tative Hopf algebra over a field (occasionally with the assumption that Stable() is monogenic or that is connected). Now we start to make use of particular properties of the Steenrod algebra. We define the dual A of the Steenrod algebra, and we give a classification of the quotient Hopf algebras of A in Section 2.1. We also define several important families of quotient Hopf algebras of A: the A(n)'s, the elementary quotients, * *and the quasi-elementary quotients. We classify the latter two families, at least a* *t the prime 2, and for each (quasi-)elementary E, we compute the homotopy groups of the ring spectrum HE. In Section 2.2 we introduce Pst-homology, a well-known tool for studying Ext over the Steenrod algebra. In our setting, Pst-homology is a homology theory, and hence is represented by an object Pstin Stable(A); we compute ss**Pst, and we perform a few other computations. Starting in Section 2.3, we begin to get to the main results. We discuss a vanishing line theorem in Section 2.3: given conditions on the Pst-homology gro* *ups of an object X, then ssijX = 0 when mi > j +c for some numbers m and c. (This is an extension to the cochain complex setting of theorems of Anderson-Davis [AD73* * ] and Miller-Wilkerson [MW81 ].) In Section 2.4 we use the vanishing line theorem to construct "self-maps of finite objects" in Stable(A). For example, if M is * *an A-comodule and dimFpM is finite, and if X is an injective resolution of M, then we construct a cochain map nX -!X which is non-nilpotent under composition. We also establish that these self-maps have certain nice properties. (This is * *an extension to the cochain complex setting of a result of the author [Pal92].) In Section 2.5 we mention a few topological applications of the vanishing li* *ne and self-map results, and one or two other issues. 2.1.Quotient Hopf algebras of A In this section we define the dual A of the mod p Steenrod algebra, we give a classification of quotient Hopf algebras of A, and we discuss two important fam* *ilies of these quotient Hopf algebras: namely, the A(n)'s and the elementary quotient* *s. Margolis' book [Mar83 ] is a good reference for all of these topics. In a subse* *ction, we also discuss the quasi-elementary quotients of A. Fix a prime number p and let A be the dual of the mod p Steenrod algebra. Recall from [Mil58] Milnor's description of A: as an algebra, we have ( A = F2[1; 2; 3; : :]:; if p = 2, Fp[1; 2; 3; : :]: E[o0; o1; o2;i:f:]:;p;is odd 23 24 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA where |n| = 2n - 1 if p = 2, |n| = 2(pn - 1) and |on| = 2pn - 1 if p odd. The coalgebra map : A -!A A is determined by Xn i : n 7-! pn-i i; i=0 Xn i : on 7-! pn-i oi+ on 1: i=0 (In both of these formulas, we take 0 to be 1.) One can check that A is a connected commutative Hopf algebra over the field Fp (or see [Mil58]); as a result, the work in the previous chapter applies to t* *he study of Stable(A). The quotient Hopf algebras of A have been classified by Anderson-Davis (p = * *2) and Adams-Margolis (p odd). We state the classification theorem here; see [Mar8* *3 ] or the original papers for the proof. (Recall that we defined conormal quotient* * Hopf algebras in Definition 1.3.7.) Theorem 2.1.1.[AD73 , AM74 ] (a)Every quotient Hopf algebra B of A is of the form ( 2n1 2n2 B = A=(1pn;12pn;2: :):;e e p = 2; A=(1 ; 2 ; : :;:o00; o11;p:o:):;dd; for some exponents n1; n2; . .2.{0; 1; 2; : :}:[ {1} and e0; e1; . .2.{1; * *2}. These exponents satisfy the following conditions: (i)For all primes p: for each i and r with 0 < i < r, either nr nr-i- i or nr ni. (ii)For p odd: if er = 1, then for each i and j with 1 i r and i + j = r, either ni< j or ej= 1. (b)Conversely, any set of exponents {ni} and {ei} satisfying conditions (i)-(* *ii) above determines a quotient Hopf algebra of A. (c)Let B be a quotient Hopf algebra of A as in (a). Then B is a conormal quotient Hopf algebra of A if and only if, for p = 2, (i)n1 n2 n3 : :,: and for p odd, (i)n1 n2 n3 : :,: (ii)e0 e1 e2 : :,: (iii)ek = 1 ) nk = 0. For part (a), if some ni= 1, that means that one does not divide out by any power of i. Similarly, if some ei= 2, then one does not mod out by oi. Part (a) says that there is a (monomorphic) function from the set of quotient Hopf algebras of A to the set of sequences either of the form (n1; n2; : :):, o* *r of the form (n1; n2; : :;:e0; e1; : :):; part (b) gives the image of this function. Fo* *r p = 2, given a Hopf algebra B, one can view the sequence of exponents n1; n2; : :a:s a function {1; 2; :-:}:!{0; 1; 2; : :}:[ {1}; i7-! ni: We refer to this as the profile function of B. There is, of course, a similar f* *unction when p is odd. We will occasionally give graphical representations of quotient * *Hopf 2.1. QUOTIENT HOPF ALGEBRAS OF A 25 | | . | 2r ..: : : | ..| n__. | r-1| 2r-1 |.. . | .: ::n::|: |p2. ..| 42 43.. |1 |22__2_ |p___ ..: : : 1 |12_|3 |1 |____. | | : : : 0 _________________________|1 __________________________|1 1 2 3: ::::n:: : : : :o:non+1: : : p = 2 __________p_odd__________|| Figure 2.1.A. Graphical representation of a quotient Hopf alge- bra of A. For p = 2, this is a barrchart; the nth column is height r - 1 if one is dividing out by 2n. For p odd, this is a similar bar chart, together with an extra row at the bottom; in this row, one marks which on's are nonzero in the quotient. algebras via their profile functions, as in [Mar83 , p. 234-5]. See Figure 2.1.* *A, for example. Here is a simple, but useful, result. See Notation 1.3.9 for the definition * *of D[x] and E[x]. Lemma 2.1.2. (a)Suppose that B is a quotient Hopf algebra of A, and that for somess and t, we have - pt6=s0+in1B, - pts = 0 in B, - pj= 0 in B for all j < t. Then there is a Hopf algebra extension s D[pt] -!B -!C: (b)Fix p odd. Suppose that B is a quotient Hopf algebra of A, and that for some n we have - on 6= 0 in B, - oj= 0 in B for all j < n, - j= 0 in B for all j n. Then there is a Hopf algebra extension E[on] -!B -!C: s Proof. For part (a), one only has to check that given the conditions on B, t* *hen ptis primitive in B (and similarly for part (b)). This check is straightforward* *. |___| Remark 2.1.3. (a)Hence the results of Lemma 1.3.10 apply. The usual notation is: s 1;|ps| hts= [pt] 2 HB1;|pst|= ExtB t(Fp; Fp); ps| bts= fifP0(hts) 2 HB2;|ppst|= Ext2;p|tB(Fp; Fp); vn = [on] 2 HB1;|on|= Ext1;|on|B(Fp; Fp): (b)Also, note that if B is a finite-dimensional quotient of A, then one can a* *lways find an integer n or a pair (s; t) so that the hypotheses of Lemma 2.1.2 h* *old. 26 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA n+1| n | | | n|@ n-1|@ | @ |@ | @ | @ | | | @ | @ | @ | @ | @ | @ _________________________|nn+1@_________________________|n@ o0 : : :on p = 2 __________p_odd__________|| Figure 2.1.B. Profile function for A(n). The actual profile func- tions have a staircase shape, which we abbreviate as lines with slope -1. At the prime 2, for instance, one can take s and t to be as follows: t = min{n | n 6= 0 inB}; i s = max{i | 2t6= 0 inB}: This provides an inductive procedure for studying finite quotients of A. We need to use several different families of quotient Hopf algebras of A; he* *re is the first family. These quotients are quite well-known; see [Mar83 , p. 235]* *, for example. Example 2.1.4.We define A(n) as follows; see also Figure 2.1.B: ( n+1 n A=(21 ; 22; : :;:2n+1; n+2; n+3; : :):; p = 2; A(n) = pn pn-1 A=(1 ; 2 ; : :;:pn; n+1; n+2; : :;:on+1; on+2;p:o:):;dd: Then A(n) is a quotient Hopf algebra of A, and the map A -!A(n) is an isomor- phism below degree |2n+11|n= 2n+1; p = 2; |p1| = 2(p - 1)pn;p odd: One important property of the A(n)'s is that the dual A* of A is the union of the duals of the A(n)'s, and this gives A* the structure of a "P-algebra" (s* *ee [Mar83 , Chapter 13] for the precise definition). In our setting, this translat* *es into the following (cf. [Mar83 , Proposition 13.4]). Proposition 2.1.5.Suppose that Y is a spectrum so that for each i, ssijY = 0 when j 0. Then Y is the sequential limit of : :-:!HA(3)^ Y -! HA(2)^ Y -! HA(1)^ Y -! HA(0)^ Y: Hence for any spectrum X, there is a Milnor exact sequence 0 -!lim-1[X; HA(n)^ Y ]i-1;j-![X; Y ]i;j-!lim-[X; HA(n)^ Y ]i;j-!0: Proof. We write the cochain complex Y as : :-:!Yj -!Yj+1 -!: :.:We may assume that for each j, the injective comodule Yj is bounded below. Because A i A(n) is an isomorphism in a range of dimensions increasing with n, the inve* *rse system of comodules : :-:!(A 2A(n)Fp) Yj-! (A 2A(n-1)Fp) Yj-! (A 2A(n-2)Fp) Yj-! : : : 2.1. QUOTIENT HOPF ALGEBRAS OF A 27 stabilizes in any given degree, and the inverse limit is Yj. We let (A 2A(n)Fp)* * Y denote the cochain complex which is (A 2A(n)Fp) Yj in degree j; then the inver* *se system of cochain complexes stabilizes in each bidegree, and the inverse limit * *is Y . This finishes the proof. (Note that (A 2A(n)Fp) Y is isomorphic to_HA(n)^ Y in Stable(A).) |__| We consider one other interesting family of quotient Hopf algebras. See [Wil* *81 ] for some results related to these sorts of Hopf algebras. Definition 2.1.6.We say that a connected commutative Hopf algebra B over a field k of characteristic p is elementary if it is isomorphicn(as a Hopf alge* *bra) to a tensor product of Hopf algebras of the forms k[x]=(xp ) with x primitive (and* * |x| even, if p is odd), and E[y] with y primitive (and |y| odd, if p is odd). (In o* *ther words, B is bicommutative and its dual B* has zp = 0 for all z in the augmentat* *ion ideal.) See [Mar83 ], [Lin] and [Wil81 ] for the following. Proposition 2.1.7. (a)Suppose that p = 2. A quotient Hopf algebra B of A is elementary if and only if it has the form n1 2n2 B = A=(21 ; 2 ; : :):; where for some r, we have (i)if i < r, then ni= 0, (ii)if i r, then ni r. Conversely, any quotient algebra B given by exponents nisatisfying (i)-(ii) is an elementary quotient Hopf algebra of A. (b)Suppose that p is odd. A quotient Hopf algebra B of A is elementary if and only if it has the form B = A=(1; 2; : :;:oe00; oe11; : :):~=E[o1-e00; o1-e11; : :]: for any exponents e0, e1, : :,:or n1 pn2 e e B = A=(p1 ; 2 ; : :;:o00; o11; : :):; where for some r, we have (i)if i < r, then ni= 0, (ii)if i r, then ni r, (iii)if i < r, then ei= 1. Conversely, any quotient algebra B given by these exponents niand eiis an elementary quotient Hopf algebra of A. We also describe the maximal elementary quotient Hopf algebras of A; see Figure 2.1.C. Every elementary quotient Hopf algebra of A is a quotient of one * *of these. Corollary 2.1.8. (a)Suppose that p = 2. The maximal elementary quo- tient Hopf algebras of A are m+1 2m+1 2m+1 E(m) = A=(1; : :;:m ; 2m+1; m+2 ; m+3 ; : :):; m 0: 28 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA | | .: : : || .. || |________________ 4|| :|: : || ..| E(4) 3|| :.:.|: 2|| .. | || :.:.: | 1||_______________________|. || E(0) | 0_________________________||| 1 2 3 4 5 6 p = 2 Figure 2.1.C. Profile functions for maximal elementary quotients of A at the prime 2. For reference,swe have included a staircase above which are the elements 2twith s t. (b)Suppose that p is odd. The maximal elementary quotient Hopf algebras of A are E(-1)0= A=(1; 2; : :):~=E[o0; o1; : :]:; m+1 pm+1 pm+1 E(m)0= A=(1; : :;:m ; pm+1; m+2 ; m+3 ; : :;:o0; : :;:om ); m 0: Note that the quotient Hopf algebras E(m) and E(m)0are conormal for all m. We use the elementary quotient Hopf algebras of A to prove the vanishing line theorem of Section 2.3; we need to know their coefficient rings. The following * *is standard. Proposition 2.1.9. (a)Let p = 2, and assume that E is an elementary quotient Hopf algebra of A. Then s HE**= F2[hts| 2t6= 0 inE]: Here, |hts| = (1; |2st|). (b)Let p be odd, and assume that E is an elementary quotient Hopf algebra of A. Then s ps HE**= E[hts| pt6= 0 inE] Fp[bts| t 6= 0 inE] Fp[vn | on 6= 0 inE]: s ps Here, |hts| = (1; |pt|), |bts| = (2; p|t |), and |vn| = (1; |on|). s ps Indeed, if E is an elementary quotient of A with pt6= 0, then t issprimitive in E. Following Notation 1.3.9 and Remark 2.1.3(a), we have hts= [pt]; similarl* *y, if on is non-zero in E, then it is primitive and we have vn = [tn]. 2.1.1. Quasi-elementary quotients of A. We also need to consider one other family of quotient Hopf algebras, the "quasi-elementary" Hopf algebras. It turns out that when p = 2, these coincide with elementary Hopf algebras; when p is odd, there are quasi-elementary Hopf algebras which are not elementary, but * *we do not have a complete classification. The quasi-elementary quotients of A arise in the nilpotence theorem 3.1.6 of Section 3.1, and indeed in many of the results of Chapters 3 and 4; the lack of* * a classification at odd primes is one obstacle to proving those theorems when p i* *s odd. So when p is 2, we have already studied these; when p is odd, we have no immedi* *ate 2.2. Pst-HOMOLOGY 29 use for them. Hence the contents of this subsection may be safely ignored, exce* *pt for the term "quasi-elementary." Definition 2.1.10.Recall from Notation 1.3.9 that we have a Steenrod oper- ation fifP0 acting on Ext, when p is odd. We say that a connected commutative Hopf algebra B over a field k is quasi-elementary if no product of the form Q Qw2Sw; Q p = 2; ( u2Soddu)( v2SevenfifP0v);p odd; is nilpotent, for any finite sets S Ext1;*B(k; k) - {0}, Sodd Ext1;oddB(k; k) * *- {0}, and Seven Ext1;evenB(k; k) - {0}. Every elementary Hopf algebra is quasi-elementary; Wilkerson [Wil81 , Section 6] gives several examples of quasi-elementary Hopf algebras which are not eleme* *n- tary. In particular, one of his examples is a quotient of A, when p is odd. S* *ee [Wil81 , Counterexample 6.3] for the following. Example 2.1.11.Suppose that p is odd, and let B be this quotient Hopf alge- bra of A: 2 p B = Fp[1; 2; 3]=(p1; p2; 3): Then B is quasi-elementary, but not elementary. When p is 2, things are a bit nicer. See [Wil81 , Theorem 6.4] for the follo* *wing. Proposition 2.1.12.Suppose that p = 2. A quotient Hopf algebra B of A is elementary if and only if it is quasi-elementary. Note that we have classified the elementary quotients of A in Proposition 2.* *1.7 and Corollary 2.1.8. We have also computed the coefficient rings of these Hopf algebras in Proposition 2.1.9. At odd primes, we do not have a classification of the quasi-elementary Hopf algebras, so we content ourselves with the following. This would follow from Co* *n- jecture 3.4.1. Conjecture 2.1.13.Suppose that p is odd. Then every quasi-elementary quo- tient Hopf algebra of A is a quotient of 2 p3 pn D = A=(p1; p2; 3 ; : :;:n ; : :):: See [Pal97] for some results relating to quasi-elementary Hopf algebras. 2.2.Pst-homology In this section we discuss Pst-homology; this tool has been used by many aut* *hors to study ExtA, so it is reasonable to expect it to be useful in the present set* *ting. [Mar83 , Section 19.1] is a good reference for basic results on Pst-homology of modules; in this section, we prove analogues of some of those results. Let A* be the dual of A. Note that an A-comodule M with structure map is naturally an A*-module, via the map A* M -1--!A* A M -ev1--!M: We remind the reader that if one dualizesswith respect to the monomial basis for A, then Pstis the element of A* dual to pt, and (when p is odd) Qn is the eleme* *nt 30 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA dual to on. If s < t, then (Pst)p = 0, so in this case one may define the Pst-h* *omology of an A-comodule M by (Pst)p-1 H(M; Pst) = ker(M_-----!_M)_Ps: im(M --!M) t For all n we have Q2n= 0, so one may define H(M; Qn) similarly. One may do these things, but that does not mean that one should do them when working in the context of cochain complexes of injective A-comodules. Definition 2.2.1.Given integers s and t with 0 s < t, we define the Pst- homology spectrum, Pst, to be the cochain complex with jth term (kp|ps t |A; j = 2k; (Pst)j= ps (kp+1)|tA|; j = 2k + 1; with differentials given as in the following diagram: : : :--! As --! A s--! A s --! A --! : : : pt 7-! 1; (p-1)pt7-! 1; pt 7-! 1 (A is a rank 1 cofree comodule, so a comodule map into A is determined by what hits 1_see Lemma 1.1.11.) We define the connective Pst-homology spectrum, pst, * *to be the cochain complex obtained by truncatingsthescomplex+for1Pstat homological dimension zero; in other words, pst= HFp[pt]=(pt ). Similarly, when p is odd, we define the Qn-homology spectrum, Qn, to be the cochain complex with jth term (Qn)j= j|on|A, and with differentials given by : : :--! A --! A --! A --! A --! : : : on 7-! 1; on 7-! 1; on 7-! 1 We define the connective Qn-homology spectrum, qn, to be the truncation of the * *Qn complex below homological degree zero; it is equal to HE[on]. For any spectrum X, we define its Pst-homology to be (Pst)**X, and we define its Qn-homology to * *be (Qn)**X. In the remainder of this section, we compute the coefficient rings of these * *spec- tra and we prove a few results for later use. s ps+1 Proposition 2.2.2. (a)If s < t, then Fp[pt]=(t ) is a quotient coalge- bra of A over which A is injective as a right comodule. For any n 0, E[on] is a quotient Hopf algebra of A (and hence A is injective as a right comod* *ule over it). (b)Hence, ( (pst)**~= F2[hts]; p = 2; Fp[bts] E[hts];p odd; s ps where |hts| = (1; |pt|) and |bts| = (2; p|t |). Also, (qn)**~=Fp[vn]; where |vn| = (1; |on|). 2.2. Pst-HOMOLOGY 31 (c)Hence ( 1 (Pst)**= F2[hts];1 p = 2; Fp[bts] E[hts];p odd; (Qn)**= Fp[v1n]; and for any connective X, ( -1 s (Pst)**X = hts(pt)**X;p-=12; bts(pst)**X;p odd; (Qn)**X = v-1n(qn)**X: Proof. Part (a) is well-known. It is dual to the statement that A*, the dual of A, is free over the subalgebra Fp[Pst]=(Pst)p; i.e., H(A*; Pst) = 0. (And si* *milarly, A* is free over E[Qn]; i.e., H(A*; Qn) = 0.) See [AM71 ], [MP72 ], or [Mar83 , Proposition 19.1]. Parts (b) and (c) are standard, trivial, or both (using Lemma 1.3.4, for in- stance). See [Mar83 , Propositions 19.2-3] for an alternate formulation_of part (c). |__| By the way, we note that for all primes, the spectrum p0t= HFp[t]=(pt) is a ring spectrum, while P0tis a field spectrum for p = 2 (perhaps P0tis an example* * of an "Artinian ring spectrum" when p is odd). Similarly, when p is odd, qn = HE[o* *n] is a ring spectrum and Qn is a field spectrum. The spectra pstfor s > 0 are not ring spectra, but are examples of the sort discussed in Proposition 1.3.2(b). Remark 2.2.3.As one might expect, there is a relationship between the mod- ule definition of Pst-homology (given just before Definition 2.2.1) and the hom* *ology functor represented by the Pst-homology spectrum: if X is an injective resoluti* *on of a comodule M, then (Pst)**X = H*(M; Pst) (Pst)**: (Here we are viewing the singly-graded vector space H*(M; Pst) as doubly-graded by putting Hi(M; Pst) in bidegree (0; i).) In particular, (Pst)**X = 0 if and o* *nly if H*(M; Pst) = 0. Similarly, we have (Qn)**X = H*(M; Qn) (Qn)**: It is convenient to have alternate notation for the spectra Pstand Qn, based on the "slopes" of the polynomial generators in their coefficient rings. ps| Notation 2.2.4.We define the slope of Pstto be p|t_2, and the slope of Qn to be |on|. Note that these spectra all have distinct slopes. The set {|2st| :s < t}; p = 2; ps| {p|t_2:s < t} [ {|on| :n 0};p odd; is called the set of slopes of A; the phrase "fix a slope n" means "fix an elem* *ent n of this set." Given a slope n, we let Z(n) denote the corresponding Pstor Qn spectrum. For example, when p = 2, we have s s Z(1) = P01; Z(3) = P02; Z(6) = P12; Z(7) = P03; : :;:Z(|2t|) = Pt; : ::: 32 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA When p is odd, we have Z(1) = Q0;Z(p - 1) = Q1; Z(p2- p) = P01; Z(p2- 1) = Q2; : :;: ps| Z(p|t__2) = Pst; : :;:Z(|om |) = Qm ; : ::: We let z(n)sbe the connective cover of Z(n). We let yn denote the element of A, either ptor om , with slope n. See [Mar83 , 19.21] for the following at the prime 2. Proposition 2.2.5.LetsB be a quotient Hopf algebra of A. Then (Pst)**HB 6= 0 if and only if pt6= 0 in B. For p odd, (Qn)**HB 6= 0 if and only if on 6= 0 i* *n B. We will use this result in Section 5.3. Proof. We will give the proof for Pstand leave the Qn proofsfor the reader. By Remark 2.2.3, we are interested in H(A 2BFp; Pst). If pt6= 0 in B, then 1 2 A 2BFpgenerates a nonzero homology class. To prove the converse, first we reduce to the case when B is finite. We defi* *ne B(n) to be the quotient Hopf algebra of A defined by the following pushout diag* *ram of Hopf algebras: A ----! B: ?? ? y ?y A(n)----! B(n) (A(n) is defined in Example 2.1.4.) In other words, B(n) is the quotient of B induced by the map A i A(n). Each B(n) is finite, and one can apply Proposi- ~= tion 2.1.5 to see that (Pst)**HB -! lim-n(Pst)**HB(n). s So it suffices to show that if B is a finite quotient of A in which pt= 0, t* *hen (Pst)**HB = 0. We do this by induction on dimFpB. The induction starts with B = Fp, in which case HFp= A. Then (Pst)**HFp= (HFp)**Pst; the homology of the cochain complex Pst. This complex is acyclic, so its homolo* *gy is zero. (Alternatively, see the remark followingsthe proof.) This starts the indu* *ction. Now fix a finite-dimensional B with pt= 0 in B, and assume that for every proper quotient C of B, we have (Pst)**HC = 0. As noted in Remark 2.1.3, we have a Hopf algebra quotient C of B with Hopf algebra kernel either Fp[pqr]=(pq+1r) * *or E[om ]. So by Lemma 1.3.10 this leads to a cofiber sequence (in which we neglect suspensions) HB f-!HB -!Z; where either Z = HC or Z is the cofiber of a self-map of HC. In either case, (Pst)**HC = 0 ) (Pst)**Z = 0, so (Pst)**f is an isomorphism. Now we argue essentially as in the proof of Lemma 2.3.11 to see that bound- edness of the comodule A 2BFpimplies that (Pst)**HB = 0. To be precise, we first have to specify the degree of the map f. There are three cases: q (1)p = 2: then f :1;|2r|HB -!HB. (2)p odd, Hopf algebra kernel E[om ]: then f :1;|om|HB -!HB. 2.2. Pst-HOMOLOGY 33 q (3)p odd, Hopf algebra kernel Fp[pqr]=(pq+1r); then f :2;p|pr|HB -!HB. We deal with cases (1) and (3); (2) is handled the same way. By our computations in Proposition 2.2.2, we know that for all i and j, we have an isomorphism (Pst)ijHB ~=(Pst)i+2;j+p|pst|HB: In cases (1) and (3), f induces an isomorphism (Pst)ijHB ~=(Pst)i+2;j+p|pqr|HB: Combining these, for each integer k we get an isomorphism (Pst)i;jHB ~=(Pst)i;j+pk(|pst|-|pqr|)HB: s q ps Now, pqr6= 0 in B and pt = 0, so pr 6= t . In particular, these two elements have different degrees. By Remark 2.2.3, for fixed i, (Pst)i;*HB is equal (up * *to suspension) to H*(A 2BFp; Pst); since A 2BFpis a bounded below comodule, then this homology must be zero in small enough degrees. By the above isomorphism,_ we may then conclude that it is zero in all degrees. |__| We remark that it is easy to show that (Pst)**A = 0_this follows by a result of Milnor-Moore [MM65 ], as in [Mar83 , Proposition 19.1]. It seems as though there should be a similar proof of Proposition 2.2.5, but we have not been able* * to find one. Recall from [HPS97 , A.2.4] that if X is any object in Stable(A), then DX denotes the Spanier-Whitehead dual of X. DX has the property that for any Y , [X ^ Y; S0] = [Y; DX]: We mention the following easy fact. Proposition 2.2.6.Let E and X be spectra with X finite. Then E**X = 0 if and only if E**DX = 0. Proof. If X is E-acyclic, then so is X ^Y for any Y . In particular, DX ^X ^* * __ DX is E-acyclic. Since DX is a retract of this when X is finite, we are done. * * |__| Corollary 2.2.7.Let X be a finite spectrum. Then (Pst)**X = 0 if and only if (Pst)**DX = 0; and for p odd, (Qn)**X = 0 if and only if (Qn)**DX = 0. One should be able to get more precise information about the relationship between (Pst)**X and (Pst)**DX, as in [Mar83 , 19.12], but we do not need it. We end this section with a note on operations on Pst-homology. The next resu* *lt follows from Remark 2.2.3. Proposition 2.2.8.We have (Pst)**Pst= H*(A 2D[pst]Fp; Pst) (Pst)**; (Qn)**Qn = H*(A 2E[on]Fp; Qn) (Qn)**: Margolis has calculated H*(A 2D[pst]Fp; Pst) for s = 0 at the prime 2 [Mar83* * , 19.26]; similar calculations work at odd primes for H*(A 2E[on]Fp; Qn). 34 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA | | i| | | | ______slope__1_oe | d-1 | | | __ __ |____ __ __ __ __ __ __ __ __ | | i|1 | ___________________________________________|||||| j0- ff j0 | j - i | | | | | | | __ __ |____ __ __ __ __ __ __ __ __ i|0 | ssijX Figure 2.3.A. Vanishing line at the prime 2. In this picture, ssijX = 0 above the slanted line (the vanishing line), to the left of the vertical line j - i = j0, and below the horizontal line i = i0. ff is a number depending only on the slope d. 2.3. Vanishing lines for homotopy groups In this section we prove several theorems relating the vanishing of Pst- and* * Qn- homology groups of an object X to the homotopy groups of X; these are versions in the cochain complex category of well-known theorems about modules over the Steenrod algebra. These results are at the heart of many of the other results o* *f the book. Recall that if X is an injective resolution of a comodule M, then ss**(X) = Ext**A(Fp; M). We make heavy use of Notation 2.2.4 in this section. CL-spectra are defined * *in Definition 1.4.6. Here are our main theorems. In the setting of modules over the Steenrod algebra, the first is due to Anderson-Davis [AD73 ] (p = 2) and Miller-Wilkerson [MW81 ] (p odd), and the second to Adams-Margolis [AM71 ] (p = 2) and Moore- Peterson [MP72 ] (p odd). Both of the results are proved in [MW81 ]; we follow those proofs. Theorem 2.3.1 (Vanishing line theorem).Let X be a CL-spectrum. Suppose that there is a number d so that Z(n)**X = 0 for all slopes n with n < d. Then ss**X has a vanishing line of slope d: for some c, we have ssijX = 0 when j < d* *i-c. Of course, this vanishing line has slope _1_d-1in (j - i; i)-coordinates. Se* *e Fig- ure 2.3.A for a picture. Theorem 2.3.2.Let X be a CL-spectrum, and assume that Z(n)**X = 0 for all slopes n; then ss**X has a "horizontal" vanishing line: for some c, we have ssijX = 0 when i > c. Corollary 2.3.3.Under the hypotheses of Theorem 2.3.2, X is in the local- izing subcategory generated by A. Proof. This follows from Lemma 1.4.8. |___| 2.3. VANISHING LINES FOR HOMOTOPY GROUPS 35 Remark 2.3.4. (a)When we speak of phenomena "above" a vanishing line, we mean "above" in the grading of Figure 2.3.A_i.e., in the region in which ssijmight be zero. (b)We will see in the proof that the "intercept" c of the vanishing line in Theorem 2.3.1 may be given by c = (d - 1)i1- j0+ ff, where ff = ff(d) is a number depending only on d. In the grading of Figure 2.3.A, the homotopy ssijis zero above the line of slope _1_d-1through the point (j0 - ff; i1).* * To compute ff, we find the smallest integer n so that the map A -! A(n) is an isomorphism through degree d (i.e., so that 2n+1 > d when p = 2, or 2(p - 1)pn > d when p is odd); then 8 X s >>> |pt|; if p = 2, >>>s+tn+1 < |ps|d ff = > Xt ps X >>> (d + (p - 1)|t |) + |oi|;if p is odd. >>:s+tn in |pst|d |oi|d For Theorem 2.3.2, the intercept is i1_the homotopy ssijis zero if i > i1. (c)For both of these theorems (in fact, for all of the results of this sectio* *n), we can weaken condition (ii) of Definition 1.4.6 slightly_it suffices to assu* *me that for all i, there is a j0 = j0(i) so that ssijX = 0 if j - i < j0. Wit* *h this assumption, one replaces j0 in the above formulas for c with min{j0(i) | i0 i i1}. Let M be a bounded below A-comodule with injective resolution X; then it is easy to see that X is CL. Furthermore, we know from [Mar83 , Theorem 13.12] that ssstX = ExtstA(Fp; M) has a "horizontal" vanishing line if and only if M is injective. Hence our theorems do indeed provide generalizations of the previous* *ly cited ones. One can generalize these results somewhat. The same proofs carry over essen- tially unchanged, so to keep the notation simple we prove Theorems 2.3.1 and 2.* *3.2, rather than the following. Theorem 2.3.5.Let X be a CL-spectrum, and fix a quotient Hopf algebra B of A. (a)Suppose that there is a number d so that Z(n)**X = 0 for all slopes n with n < d and yn 6= 0 in B. Then HB**X has a vanishing line of slope d: for some c, we have HBijX = 0 when j < di - c. (b)If Z(n)**X = 0 for all slopes n with yn 6= 0 in B, then HB**X has a "horizontal" vanishing line; hence HB ^ X is in the localizing subcategory generated by A. Of course, HB** relates to ExtB the same way ss** relates to ExtA, so one should view this result as a vanishing line theorem over B. (One could also work in Stable(B) and prove a result about ss**in that category, but it seems better* * to work in Stable(A) whenever possible.) Remark 2.3.4 also applies here. 2.3.1. Proof of Theorems 2.3.1 and 2.3.2 for p = 2. We give the proofs of Theorems 2.3.1 and 2.3.2 when p = 2. In the next subsection we indicate the changes necessary when working at odd primes. 36 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA Recall from Lemmas 1.3.10 and 2.1.2, as well as Remark 2.1.3, that if B is a finite-dimensional quotient Hopf algebra of A, then for some s and t we have the following conditions: o2st6= 0 in B, o2s+1t=s0 in B, o2j= 0 in B for all j < t. Hence there is a Hopf algebra extension s E[2t] -!B -!C; hence an element hts2 HB1;|2st|and a cofibration 2s| hts 1;|tHB --! HB -!HC: Given this situation, we have the following result. (Given a spectrum Y and a self-map v :ijY -! Y , we say that v acts nilpotently on ss**Y if for all y 2 s* *s**Y , some power of ss**(v) annihilates y.) Lemma 2.3.6.Assume that B is finite-dimensional, and fix s and t as above. (a)If s t, then hts2 HB**is nilpotent; hence the self-map hts:HB -!HB is nilpotent. (b)Given an object X, consider the cofibration 2s| hts^1X 1;|tHB ^ X -----!HB ^ X -!HC ^ X: If hts^ 1X acts nilpotently on HB**X, and if HC**X has a vanishing line of slope d, then HB ^ X has a vanishingsline of slope d. The difference in intercepts depends only on d and |2t|_it is independent of X. Remark 2.3.7.In fact, the nilpotence of htsdoes not depend on B being finite-dimensional. See Theorem B.2.1 for a generalization, due to Lin. Proof. Lin proved part (a) in [Lin, Corollary 3.2]; see also Anderson-Davis [AD73 ] and Miller-Wilkerson [MW81 , Proposition 4.1]. For part (b), we look at the long exact sequence in homotopy coming from the given cofibration (we write h for ss**(hts^ 1X )): : :-:!HCi-1;jX -!HBi-1;j-|2st|X h-!HBijX -!HCijX -!: ::: Fix a bidegree (i; j)sabove the vanishing line for HC**X. There are two cases: suppose first that |2t| > d. The exact sequence tells us that HBi-1;j-|2st|X h-!HBijX is an epimorphism. Since |2st| > d, then (i - 1; j - |2st|) is above the vanish* *ing line for HC**X. Hence HBi-k;j-k|2st|X h-!HBi-(k-1);j-(k-1)|2st|X is an epimorphism for all ks> 0. Also since |2st| > d, then one can see that for k 0, (i - k; j - (k - 1)|2t|) is a bidegree above the vanishing line for HC**X* *, so HBi-k;j-k|2st|X h-!HBi-(k-1);j-(k-1)|2t|X is an isomorphism. Since h acts nilpotently on HB**X, though, these groups must be zero for k large; since they surject onto HBijX, then HBijX = 0. Note that in this case, the vanishing line for HC**X is the same as that for HB**X. 2.3. VANISHING LINES FOR HOMOTOPY GROUPS 37 Suppose, on the other hand, that |2st| d. If (i; j) is a bidegree above the vanishing line, then so is (i + 1; j), so the map HBi;j-|2st|X h-!HBi+1;jX is an isomorphism. Arguing as above, we see that the group HBi;j-|2st|X must be zero. In this case, the interceptsof the HB vanishing line changes_by |2st|: HBijX = 0 when j < di - c - |2t|. |__| Lemma 2.3.8.Fix an object X satisfying condition (i) of Definition 1.4.6. (a)Given an extension s E[2t] -!B -!C where B and C are quotients of A, if |2st| d and if HC**X has a vanishing line of slope d, then HB**X has a vanishing line of slope d. In fact, HB**X has the same vanishing line as HC**X. (b)Given finite-dimensional quotients B i C of A, if HC**X has a vanishing line of slope d and if the map B i C is an isomorphism in dimensions less than d, then HB**X has a vanishing line of slope d. In fact, HB**X has the same vanishing line as HC**X. Proof. Part (a): (This proof is based on that of [MW81 , Proposition 3.2].) We assume that HCi;jX = 0 when j < di-c, and we want to show that HBi;jX = 0 when j < di - c. We prove this by induction on i. By Lemma 1.3.10, we have a cofibration 2s| hts 1;|tHB ^ X --! HB ^ X -!HC ^ X: This gives us a long exact sequence in homotopy: (2.3.9) : :-:!HBi-1;j-|2st|X -!HBi;jX -!HCi;jX -!: ::: Since X satisfies condition (i) of Definition 1.4.6, then for i sufficiently sm* *all and for all j, we have ssi;j(X) = 0 = HBi;jX = HCi;jX: This starts the induction: if i0is the smallest value of i for which HCi;jX is * *nonzero, then we have an inclusion HBi0;jX ,! HCi0;jX (for all j). Thesinductive step is also easy: if we have i and j with j < di - c, then j - |2t| < d(i - 1) - c; hence both HCi;jX and HBi-i;j-|2st|X are zero. So we apply exactness in the long exact sequence (2.3.9). Part (b): Given B i C with B finite-dimensional (or more generally, with C of finite index in B), then there is a sequence of extensions s1 E[2t1] -!B -!B1; s2 E[2t2] -!B1 -!B2; .. . sn E[2tn] -!Bn-1 -!C: (See [HSb , Lemma A.11] or [MW81 , Lemmas 3.4-3.5], for example.)sIf, further- more, B i C is an isomorphism through degree d - 1, then each 2tihais degree_ at least d. So apply induction and part (a). |__| 38 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA Lemma 2.3.8 is a special case of the following; we have stated and proved th* *em separately because at odd primes we need a variant of the former, while the lat* *ter holds as stated. Lemma 2.3.10.Fix an object X satisfying condition (i) of Definition 1.4.6. Given a surjection of coalgebras f :A i B, if f is an isomorphism below degree * *d, and if HB**X has a vanishing line of slope d, then HA**X has a vanishing line of slope d. In fact, they have the same vanishing line. Proof. We have an injection of comodules A 2AF2-! A 2BF2, and the cok- ernel is zero below dimension d. Taking injective resolutions gives a cofibrati* *on 1;0Z -!HA -!HB -!Z; where Z is (0; d)-connective (Definition 1.4.3). So to show that HAijX = ssijX * *=_0_ if j < di - c, one argues by induction on i just as in Lemma 2.3.8(a). * *|__| Lemma 2.3.11.Let X be an object satisfying conditions (ii) and (iii) of Defi* *ni- tion 1.4.6. Suppose that E is a finite-dimensionalselementary quotient Hopf alg* *ebra of A. If (Pst)**X = 0 whenever 2t 6= 0 in E, then HE**X has a horizontal vanishing line. In fact, given i1 as in condition (iii), then HEi*X = 0 if i > * *i1. Proof. Since the dual of E is an exterior algebra, we abuse notation and wri* *te E = E[x1;sx2; : :;:xn] where each xk is primitive; in other words, each xk is e* *qual to 2tfor some s and t. We point out that the xk's have distinct degrees. Now HE**is a polynomial algebra on n generators; we write vk 2 HE1;|xk|for the generator corresponding to xk. We also use vk to denote the corresponding s* *elf- map of HE, and we let HE=(vk) be its cofiber. Note that the self-map vk induces multiplication by vk on homotopy, so that ss**(HE=(vk)) = (ss**HE)=(vk): We define HE=(vk1; : :;:vkm) similarly (assuming that the numbers k1; : :;:km a* *re distinct). Hence if ` is an integer with 1 ` n and ` 62 {k1; : :;:km }, then v`: HE -!HE induces a self-mapsof HE=(vk1; : :;:vkm). Also note that, if xk = 2t, then by a change-of-coalgebras, we have (Pst)**X = v-1k(HE=(v1; : :;:^vk; : :;:vn))**X: We refer to this as the xk-homology of X. We claim for any set {k1; : :;:km } {1; : :;:n}, (HE=(vk1; : :;:vkm))**(X) has a horizontal vanishing line. We prove this by induction on n - m. (We will have proved the lemma when n - m = n.) We have to deal with the first few cases before we can apply the inductive s* *tep. If n - m = 0, then {k1; : :;:km } = {1; : :;:n}, so HE=(vk1; : :;:vkm) = HF2, a* *nd by assumption, HF2**X has a horizontal vanishing line with intercept i1. If n-m = * *1, then we let k be the integer so that {k} [ {k1; : :;:km } = {1; 2; : :;:n}: To simplify the notation, we let HEm = HE=(vk1; : :;:vkm). Then we have a cofi- bration HEm ^ X vk-!HEm ^ X -!HF2^ X: 2.3. VANISHING LINES FOR HOMOTOPY GROUPS 39 By hypothesis, HF2**X has a horizontal vanishing line, so the map vk induces an isomorphism in ssijfor i > i1. On the other hand, the xk-homology of X is zero,* * and the xk-homology of X is equal to v-1k(HEm )**X. Since vk induces an isomorphism for i > i1, and since this localization is zero, then vk must be zero when i > * *i1. So (HEm )**X has a horizontal vanishing line with intercept i1. Now fix m with n - m 2, and assume that (HE=(v`1; : :;:v`t))**X has a horizontal vanishing line whenever n - t < n - m. As above, let HEm = HE=(vk1; : :;:vkm). Pick distinct integers k; ` n which are not in {k1; : :;:k* *m }, and consider the following diagram (in which each row and column is a cofibrati* *on): 2;|xk|+|x`|HEm-vk---!1;|x`|HEm----!1;|x`|HEm =(vk) ? ? ? v`?y v`?y ?y 1;|xk|HEm --vk--! HEm ----! HEm =(vk) ?? ? ? y ?y ?y 1;|xk|HEm =(v`)----!HEm =(v`)----! HEm =(vk; v`) Now smash this diagram with X. By induction, all of the spectra HEm =(vk; v`)^X, HEm =(vk)^X, and HEm =(v`)^X have horizontal vanishing lines with intercept i1. Now we apply ssij(-); for i > i1, the maps labeled vk and v` induce isomorphisms on ssij, so we have v-1`Ovk ssi;jHEm ^ X -------!~ssi;j+|xk|-|x`|HEm ^ X: = This isomorphism, combined with the facts that |xk| 6= |x`| and that ssij(HEm ^* *X) is zero when j 0, implies that ssij(HEm ^ X) = 0 for all j; i.e., ss**(HEm ^ X) has the predicted horizontal vanishing line. This completes the inductive_step,* * and hence the proof. |__| Proof of Theorem 2.3.1.By Lemma 2.3.10, if we know that HB**X has a vanishing line of slope d for all finite-dimensional quotient Hopf algebras B o* *f A, then ss**(X) will, also. (For example, the map A -!A(n) is an isomorphism throu* *gh degree 2n+1- 1, so apply the lemma to the case B = A(n) where 2n+1- 1 d.) Now we show that HB**X has a vanishing line of slope d for all quotients B of A with dimF2B < 1, by induction on dimF2B: The case where dimF2B = 1 (i.e., B = F2) is taken care of by condition (iii), so we move on to the inductive ste* *p. By Remark 2.1.3, there is a Hopf algebra extension s E[2t] -!B -!C: By hypothesis, HC**X has a vanishing line of slope d, and we want to produce a vanishing line for HB**X. There are several cases: (1) If s t, then wesare done by Lemma 2.3.6. (2) If s < t and |2t| > d, then we are done by Lemma 2.3.8(a). 40 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA (3) If s < t and |2st| d, then we have to prove something. We may assume that j= 0 in B for j < t. Let s+1 2n1 2n2 2n3 B = F2[t; t+1; t+2; : :]:=(2t ; t+1; t+2; t+3; : :):; s+1 2m1 2m2 2m3 E = F2[t; t+1; t+2; : :]:=(2t ; t+1; t+2; t+3; : :):; where mi = max (min(ni; s - i); 0). By Proposition 2.1.7,uE is an elementaryu quotientsHopf algebra of A (and of B). Furthermore,uif 2v6= 0 in E, then |2v| |2t|. By hypothesis on X, then, (Puv)**X = 0 when 2v 6= 0 in E. So we apply Lemma 2.3.11 to conclude that HE**X has a horizontal vanishing line with interc* *ept i1. Since X satisfies conditions(ii) of Definition 1.4.6, then HE**X also have* * a vanishing line of slope |2t| + 1 (the line with this slope through the point (i* *1; j0)). We then apply Lemma 2.3.8(b)sto the case B i E to conclude that HB**X has a vanishing line of slope |2t| + 1 (andsin fact the same vanishing line). Now, the map htsacts at slope |2t|, and hence acts nilpotently on HB**X. By Lemma 2.3.6(b), then, the vanishing line for HC**X, which has slope_d, gives one for HB**X. |__| Proof of Theorem 2.3.2.By Proposition 2.1.5, it suffices to show that there is a uniform horizontal vanishing line for all of the groups HA(n)**X. Lemma 2.* *3.11 and Lemma 2.3.8 together show that each HA(n)**X has a horizontal vanishing line, and in fact these results identify the vanishing line: HA(n)i*X = 0 if i_* *>_i1, with i1 as in condition (iii) of Definition 1.4.6. |_* *_| 2.3.2. Changes necessary when p is odd. The above proofs of Theo- rems 2.3.1 and 2.3.2 go through with a few changes when the prime is odd; we indicate those changes in this subsection. We start with the same set-up as for the prime 2: we have an extension of Ho* *pf algebras in one of the following forms: (E) E[on] -!B -!C; s (D ) D[pt] -!B -!C: If case (E) arises, then Lemma 2.3.6(b) carries over as stated (writing vn for * *the homotopy element associated to the element on, rather than hts). Otherwise extension (D ) arises, and we may assume that we have s opt6=s0+in1B, opts = 0 in B, opj= 0 in B for all j < t. Lemma 1.3.10 then gives an element bts2 HB2;p|pst|, and hence a cofibration ps| bts 1;p|ps| (2.3.12) 2;p|tHB --!HB -!gHC -! tHB; where gHCis the cofiber of a self-map of HC, as in Lemma 1.3.10. Hence if HC**X has a vanishing line of slope d, then so does gHC**X. More precisely,sif HCijX * *= 0 when j < di - c, then gHCijX = 0 when j < di - c + min(0; |pt| - d). Here is the odd prime analogue of Lemma 2.3.6. Lemma 2.3.13. (a)Given s and t as above so that we have extension (D ), if s t, then bts2 HB**is nilpotent; hence the self-map bts:HB -!HB is nilpotent. 2.3. VANISHING LINES FOR HOMOTOPY GROUPS 41 (b)Given extension (D ) and an object X, consider the cofibration ps| bts^1X 2;p|tHB ^ X ----! HB ^ X -!gHC^ X: If bts^ 1X acts nilpotently on HB**X, and if gHC**X has a vanishing line of slope d, then HB ^ X has a vanishingsline of slope d. The difference in intercepts depends only on d and |pt|_it is independent of X. Proof. For part (a), see [MW81 , Proposition 4.1]. __ Part (b) is proved just as in the p = 2 case. |__| The odd prime version of Lemma 2.3.8 is as follows. Lemma 2.3.14.Fix an object X satisfying condition (i) of Definition 1.4.6. (a)Given an extension s D[pt] -!B -!C; 2s| if p|t_2 d and if HC**X has a vanishing line of slope d, then HB**X has a vanishing line of slope d. In fact, HB**X has the same vanishing line as HgC**X. (b)Given an extension E[on] -!B -!C; if |on| d and if HC**X has a vanishing line of slope d, then HB**X has a vanishing line of slope d. In fact, HB**X has the same vanishing line as HC**X. (c)Given finite-dimensional quotients B i C of A, if HC**X has a vanishing line of slope d and if the map B i C is an isomorphism in odd dimensions less than d and in even dimensions less than 2d_p, then HB**X has a vanish* *ing line of slope d. Furthermore, the difference in intercept between the two vanishing lines is independent of X. Proof. Part (a) is proved just as is Lemma 2.3.8(a), but based on the cofibr* *a- tion (2.3.12). Part (b) is the same as Lemma 2.3.8(a) (except for the character* *istic of the ground field, which is not relevant). Part (c) is proved, as in Lemma 2.* *3.8(b), by induction and parts (a) and (b). Since in (a), the vanishing line for gHC**X* * may have a different intercept than that for HC**X, the intercept for HB**X will ch* *ange_ as the induction proceeds, but it will change by amounts dependent only on d. * *|__| Lemma 2.3.10 holds as stated, regardless of the prime involved. Here is the analogue of Lemma 2.3.11. Lemma 2.3.15.Let X be an object satisfying conditions (ii) and (iii) of Defi* *ni- tion 1.4.6. Suppose that E is a finite-dimensionalselementary quotient Hopf alg* *ebra of A. If (Pst)**X = 0 whenever 2t6= 0 in E, and (Qn)**X = 0 whenever on 6= 0 in E, then HE**X has a horizontal vanishing line. In fact, given i1 as in condi* *tion (iii), then HEi*X = 0 if i > i1. The proof is the same as that for Lemma 2.3.11, using cofibrations of the fo* *rm ps| (2.3.12)repeatedly. We also need to point out that the slopes p|t_2and |on| are* * all distinct. (Note also that in this case, we may choose d as large as we like, so* * that a horizontal vanishing line for HC**X induces the same one for gHC**X.) 42 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA Finally, given these modified tools, the proofs of the main theorems go thro* *ugh essentially unchanged. 2.4.Self-maps via vanishing lines In this section we use vanishing lines to construct self-maps of finite spec* *tra. For each finite X, we construct one self-map; we give many others in Theorem 4.* *1.3. We make heavy use of Notation 2.2.4 in this section. Our approach for this section is based on some recent work in ordinary stable homotopy theory, as described in work of Ravenel [Rav86 ] and Hopkins and Smith [HSb ]. For now, we view the objects Pstand Qn (i.e., the Z(n)'s) as the analog* *ues of Morava K-theories. This is not a perfect analogy, because the Morava K-theor* *ies detect nilpotence, while the Z(n)'s do not. The following definition is based on the definition of "vn-map" in ordinary stable homotopy theory; see [HSb ] and [Rav92 ]. We will investigate and genera* *lize this definition in Section 4.2. Notation 2.4.1.Fix a slope n. The ring z(n)** has a polynomial subalge- bra; we call the polynomial generator un. (In the notation of Remark 2.1.3 and Proposition 2.2.2, un is one of hts, bts, or vn.) Definition 2.4.2.Fix a spectrum X and a slope n. (a)A self-map f 2 [X; X]**is a un-map if for some j, Z(n)**f = ujn^ 1X : (b)We say that X is of type n if Z(d)**X = 0 for d < n, and Z(n)**X 6= 0. Notice that if Z(n)**X = 0, then the zero map of X is a un-map. We point out that in the ordinary stable homotopy category, Ravenel showed in [Rav84 , 2.11] that for any finite spectrum X, K(n)*X 6= 0 ) K(n + 1)*X 6= 0. The analogous statement here, with Z(n)**rather than K(n)*, does not hold. See Proposition 4.8.1 (and also [Pal96b, Prop. 3.10 and Thm. A.1]) for the correct statement when p = 2 (and for a guess in the odd prime case). We do know that if X is a nonzero finite spectrum, then by Theorem 2.3.2 and Corollary 4.5.7, f* *or some n we have Z(n)**X 6= 0. (One could also use the Atiyah-Hirzebruch spectral sequence to show this.) The following first appeared (for modules) in [Pal92]. It is a slight genera* *l- ization of a result of Hopkins and Smith [HSb ]. See Theorems 3.1.2 and 4.1.3 f* *or stronger results when p = 2. Theorem 2.4.3.Fix a finite spectrum X of type n. Then for some k, there is a non-nilpotent un-map v :k;knX -!X: To prove this, we need a "relative vanishing line" result; this is a general* *ization of a standard result_see [Rav86 , 3.4.9], for instance. Lemma 2.4.4.Fix a slope n. Suppose that X is a CL spectrum, and suppose that X is of type at least n (hence ss**X has a vanishing line of slope n). Giv* *en m 0, let M be the number below which degree h: A -!A(m) is an isomorphism. Then the Hurewicz map h: ss**X -! HA(m)**X is an isomorphism above a line of slope n: for some c independent of m, h is an isomorphism on ssijwhen j < ni + M - c. 2.4. SELF-MAPS VIA VANISHING LINES 43 Proof. This is a consequence of the form of the intercept in the vanishing l* *ine theorem_see Remark 2.3.4. Let W denote the fiber of the map S0 -! HA(m). Since the kernel of the map A -! A(m) is zero below degree M, if we choose numbers i0, i1, and j0 so that X satisfies Definition 1.4.6, then W ^ X satisfi* *es_the conditions with the numbers i0, i1, and j0+ M. |__| We have the following result based on work of Lin [Lin] and Wilkerson [Wil81* * ]; this is a corollary of Theorem 3.3.5. Proposition 2.4.5.Fix m and consider the quotient Hopf algebra A(m) of A. (a)Let p = 2. Fix integers s < t with 2stnonzero in A(m) (i.e., with s + t m + 1). Then the restriction map Ext**A(m)(F2; F2) -!Ext**F2[t]=(2s+1t)(F2; F2) ~=F2[ht0; ht1; : :;:hts] is surjective modulo nilpotence (i.e., the algebra cokernel consists entir* *ely of nilpotent elements). Hence for some i = i(m), there is a non-nilpotent element 2s| w 2 Exti;i|tA(m)(F2; F2) = HA(m)i;i|2st| which restricts to hits. s (b)Let p be odd. Fix integers s < t with ptnonzero in A(m) (i.e., with s + t m). Then the restriction map Ext**A(m)(Fp;-Fp)---! Ext**Fp[t]=(ps+1(Fp; Fp) fl t ) flfl Fp[bt0; : :;:bts] E[ht0; : :;:hts] is surjective modulo nilpotence. Hence for some j = j(m), there is a non- nilpotent element ps| w 2 Ext2j;jp|tA(m)(Fp; Fp) = HA(m)2j;jp|pst| which restricts to bjts. (c)Let p be odd. Fix an integer t with ot nonzero in A(m) (i.e., with t m). Then the restriction map Ext**A(m)(Fp; Fp) -!Ext**E[ot](Fp; Fp) ~=Fp[vt] is surjective mod nilpotence. Hence for some k = k(m), there is a non- nilpotent element w 2 Extk;k|ot|A(m)(Fp; Fp) = HA(m)k;k|ot| which restricts to vkt. We need the following lemma. Recall from Notation 1.3.9 that D[y] is the Hopf algebra Fp[y]=(yp) with y primitive. Lemma 2.4.6.Fix a slope n, and choose an integer m large enough that yn is nonzero in A(m) (and hence so that D[yn] or E[yn] is a quotient coalgebra of A(* *m) over which A(m) is injective). (a)Then the map HA(m)** -!z(n)** is an algebra map, and some power of the polynomial generator un of z(n)**is in the image. 44 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA (b)For any object X, the following diagram commutes. [X; HA(m) ^ X]**----! [X; z(n) ^ X]** x x -^X?? ??-^X HA(m)** ----! z(n)** (c)For any object X with Z(n)**X 6= 0, the map z(n)**-! [X; z(n) ^ X]**is an injection. Proof. Part (a) follows from Proposition 1.3.2(b) and Proposition 2.4.5. Part (b) follows from the fact that [X; z(n) ^ X]** is isomorphic to cochain homotopy classes of D[yn]-linear (resp., E[yn]-linear) self-maps of X, and the * *hor- izontal maps are just restriction. For part (c), we merely note that if Z(n)**X 6= 0, then the identity map on * *X_ is not cochain homotopic to zero over D[yn] (resp., over E[yn]). |* *__| Proof of Theorem 2.4.3.By Lemma 2.4.4, if we choose m large enough, then the map [X; X]** -![X; HA(m) ^ X]** is an isomorphism in the bidegrees (k; kn), for all integers k. Now we apply Lemma 2.4.6 to find a non-nilpotent element w 2 HA(m)k;kn(for some k) which maps nontrivially to z(n)** and to [X; z(n) ^ X]**. The lift of w ^ 1X to [X; X]k;knclearly has the_desired_proper* *ties. |__| We will need the following lemma in Section 4.7. Lemma 2.4.7.Fix a finite type n spectrum X. Then for some k, the un-map v 2 [X; X]** constructed in Theorem 2.4.3 is central in a band parallel to the vanishing line. This band includes the origin. Proof. Consider the element w 2 HA(m)k;kn, as given by Proposition 2.4.5. Since HA(m)**is a commutative ring, w maps to a central element in [X; HA(m) ^ X]**. By Lemma 2.4.4, [X; X]**-! [X; HA(m) ^ X]**is an isomorphism in a band parallel to the vanishing line; by choice of m, that band includes the origin._* *Hence the lift of w to [X; X]**is central in that band. |__| 2.5.Further discussion As mentioned in Subsection 2.1.1, one of the main gaps in this theory, when p is odd, is the lack of a classification of the quasi-elementary quotient Hopf a* *lgebras of A. See Appendix B.3 for a discussion of conjectures and results related to t* *his issue. We note that the vanishing line theorem 2.3.1 has been used many times in many papers; it provides a convenient way to get valuable information about the Adams E2-term. Combined with newer results, such as Theorem A.2.6, it is even more powerful. Theorem 2.4.3, the result that ensures the existence of a non-nilpotent self* *-map of any finite object, also has been used in topological applications. For examp* *le, Hopkins and Smith used it to prove the periodicity theorem [HSb ]: they had constructed a particular spectrum X, and they used the theorem to find a vn-map of X at the E2-term of the Adams spectral sequence. Later, Theorem 2.4.3 was used by Sadofsky and the author in [PS94 ] to give a new proof of the periodici* *ty 2.5. FURTHER DISCUSSION 45 theorem: we used it not only to produce a vn-map, but also to construct the spectrum X in question. This was done by taking iterated cofibers: when p = 2, the cofiber of u1: S0 -!S0 has a u3-map (3 is the next slope after 1); the cofi* *ber of this map has a u6-map, etc. This was done with modules over the Steenrod algebra, and then realized at the spectrum level in the ordinary stable homotopy category. See the results of Section 4.7 for a similar application of Theorem 2.4.3, b* *ut in Stable(A). 46 2. BASIC PROPERTIES OF THE STEENROD ALGEBRA CHAPTER 3 Quillen stratification and nilpotence The vanishing line theorem of Section 2.3 is a nilpotence theorem of a sort:* * if X is a finite spectrum, then any self-map of X with slope smaller than that of * *the vanishing line of X must be nilpotent (because some power of it will lie above * *the vanishing line). We used the vanishing line theorem in Section 2.4 to construct* * a non-nilpotent self-map of any finite spectrum. In this chapter we give some related, but stronger, results; we work mostly * *at the prime 2. We let Q denote the category of quasi-elementary quotients of A, with morphisms given by quotient maps. We can assemble the individual Hurewicz (restriction) maps S0 -!HE into S0 -!lim-QHE: Since the maximal quasi-elementary quotients are conormal, there is an action of A on lim-QHE**; we prove that ss**S0 -!(lim-QHE**)A is an F-isomorphism_its kernel consists of nilpotent elements, and some pth pow* *er of every element in the target is actually in the image. One can view this as an analogue of the Quillen stratification theorem [Qui71 , 6.2], which identifies * *ss**S0 up to F-isomorphism in the category Stable(kG*), for G a finite group and k an algebraically closed field of characteristic p. Similarly, suppose that we write D for the following quotient of A: 2 p3 pn D = A=(p1; p2; 3 ; : :;:n ; : :):: Then D is conormal, and the Hurewicz map induces an F-isomorphism ss**S0 -!HDA**: These F-isomorphism theorems first appeared in [Pal]. We have weaker results with nontrivial coefficients: if R is a ring spectrum, then an element ff 2 ss**R is nilpotent if its image under the Hurewicz map in HD**R is zero, or if its image in HE**R is zero for all quasi-elementary quotie* *nts E of A. (These nilpotence theorems are improvements on results of the author in [Pal96a].) See also Theorem 4.2.2 for a statement about lifting invariants from HD**R to ss**R. We state these results more precisely in Section 3.1. In Section 3.2, we pro* *ve the two theorems_nilpotence and F-isomorphism_involving D; we then use these these to prove the analogous results for quasi-elementary Hopf algebras in Sec- tion 3.3. We end the chapter with Sections 3.4 and 3.5; in the first of these, * *we indicate why our proofs only work at the prime 2; this leads us to a conjecture about the nilpotence of particular classes in Ext over quotients of the dual of* * the 47 48 3. QUILLEN STRATIFICATION AND NILPOTENCE | | | ___ | ___ | ___| | ___| | | | ___| | ___| | | | ___| | ___| |__| |__| | | _________________________| __________________________| p = 2 __________p_odd__________| Figure 3.1.A. Profile function for D. odd primary Steenrod algebra. In the second of these, we discuss a few possible generalizations of the main results of this chapter. Except for Sections 3.4 and 3.5, we work at the prime 2 in this chapter. 3.1.Statements of theorems Let p = 2. In this section we present two F-isomorphism results and two nilpotence resu* *lts. Let D be the following quotient Hopf algebra of A: 2 p3 pn D = A=(p1; p2; 3 ; : :;:n ; : :):: See Figure 3.1.A. Note that D and A have the same quasi-elementary quotients_ i.e., every quasi-elementary quotient map A -! E factors as A -! D -! E. As a result, it turns out that HD plays a similar role in Stable(A) to that of BP in* * the ordinary stable homotopy category, at least as far as detection of nilpotence g* *oes. We write jD :S0 = HA -! HD for the unit map of the ring spectrum HD, and similarly for jE :S0 -!HE for E quasi-elementary. 3.1.1. Quillen stratification. We state our results describing ss**S0 up to F-isomorphism. The results in this subsection first appeared in [Pal]. Definition 3.1.1. (a)Given a Hopf algebra B and a B-comodule M, we define the B-invariants of M to be MB = HomB (Fp; M) M: (This is the same as the primitives PM of M: if :M -! B M is the coaction map, then we let PM = {m 2 M | (m) = 1 m}.) (b)Following Quillen [Qui71 ], if ': R -! S is a map of graded commutative Fp-algebras, we say that ' is an F-isomorphism if it satisfies the followi* *ng properties. p____ (i)Every x 2 ker' is nilpotent. (Hence ker' is the nilradicalnof R.) (ii)For any element y 2 S, there is an integer n so that yp 2 im'. In (ii), if one can choose the same n for every y, then we say that ' is a uniform F-isomorphism. Here is our first result. Theorem 2.1.1 tells us that D is conormal, so by Remark 1.3.8, there is a coaction of A 2D F2on HD**. 3.1. STATEMENTS OF THEOREMS 49 Theorem 3.1.2 (Quillen stratification,TI).he Hurewicz map ss**S0 -!HD** factors through ': ss**S0 -!HDA2DF2**; and ' is an F-isomorphism. Here is our second result, an analogue of Quillen's theorem [Qui71 , 6.2], w* *hich identifies group cohomology up to F-isomorphism. We have defined in Defini- tion 2.1.10 (see also Proposition 2.1.12) the notion of a quasi-elementary quot* *ient Hopf algebra of A; we let Q denote the category of quasi-elementary quotients of A, with morphisms given by quotient maps. In Section 3.3, we construct a coacti* *on of A on lim-QHE**. Theorem 3.1.3 (Quillen stratification,TII).he map ss**S0 -!lim-QHE**fac- tors through i jA fl :ss**S0 -! lim-QHE** ; and fl is an F-isomorphism. We prove Theorem 3.1.2 in Section 3.2; we show that Theorem 3.1.3 follows from Theorem 3.1.2 in Section 3.3. Remark 3.1.4. (a)One can view Theorem 3.1.3, and hence Theorem 3.1.2, as giving an analogue of the Quillen stratification theorem [Qui71 , 6.2]: given a finite group G and an algebraically closed field k of characteris- tic p, we let A be the category whose objects are the elementary abelian p-subgroups of G, and whose morphisms are generated by inclusions and compositions. Then the natural map H*(G; k) -!lim-E2AH*(E; k) is an F-isomorphism. The role of the conjugation maps in the category A is played, in our results, by the process of taking invariants. (b)We have an explicit formula for the algebra lim-HE** (Proposition 4.4.1), as well as for the coaction of A on lim-HE**(Proposition 4.4.4). One can use this to predict the presence of large families of non-nilpotent elemen* *ts in ss**S0. See Section 4.4 for details. (c)We do not expect the F-isomorphisms in these results to be uniform, al- though we do not have much evidence either way. (d)One can also view Theorem 3.1.2 as an analogue of Nishida's theorem [Nis73]_in the ordinary stable homotopy category, ss*S0 is isomorphic to Z, mod nilpotence; however, while Nishida's theorem does not immediately lead to guesses as to further structure in the ordinary stable homotopy categor* *y, our analogue does. For instance, see Section 4.6 for a suggested classific* *ation of thick subcategories of finite spectra in Stable(A). (e)Of course, if A0is any quotient Hopf algebra0of A, then similar results ho* *ld: for instance, the map HA0**-!(HD0**)A is an F-isomorphism. The proofs for Theorems 3.1.3 and 3.1.2 carry over easily. 50 3. QUILLEN STRATIFICATION AND NILPOTENCE 3.1.2. Nilpotence. We move on to our nilpotence theorems. The first of these is based on the nilpotence theorem of Devinatz, Hopkins, and Smith [DHS88 ]: t* *he ring spectrum BP detects nilpotence. Theorem 3.1.5 (Nilpotence theorem,TI).he ring spectrum HD detects nilpo- tence: (a)Fix a ring spectrum R and ff 2 ss**R. Then ff is nilpotent if and only if jD ^ ff 2 HD**R is nilpotent. (b)Fix a finite spectrum Y and a self-map f :Y -! Y . Then f 2 [Y; Y ]** is nilpotent if and only if jD ^ f 2 [Y; HD ^ Y ]**is nilpotent. (c)Fix a finite spectrum F, an arbitrary spectrum X, and a map f :F -! X. Then f 2 [F; X]**is smash-nilpotent if jD ^ f 2 [F; HD ^ X]**is zero. The corresponding result for modules is [Pal96a, Theorems 3.1 and 4.2]. In [Pal96a] we prove this in enough generality so that the proof goes through here without difficulty. We also give a (slightly different) proof in Section 3.2. The second nilpotence theorem is, more or less, an analogue of the K(n) nilp* *o- tence theorem in [HSb ]. The following appeared for bounded below modules in [Pal96a, Theorems 1.1 and 4.3]. Theorem 3.1.6 (Nilpotence theorem,TII).he collection of ring spectra {HE | E quasi-elementary} detects nilpotence: (a)Fix a ring spectrum R and ff 2 ss**R. Then ff is nilpotent if and only if jE ^ ff 2 HE**R is nilpotent for all quasi-elementary quotients E of A. (b)Fix a finite spectrum Y and a self-map f :Y -! Y . Then f is nilpotent if and only if jE ^ f is nilpotent for all quasi-elementary quotients E of A. (c)Fix a finite spectrum F, an arbitrary spectrum X, and a map f :F -! X. Then f is smash-nilpotent if jE ^ f :F -! HE ^ X is zero, for all quasi- elementary quotients E of A. The proof of the analogous result in [Pal96a] does not apply to nonconnective situations, so we give a new proof below. Remark 3.1.7. (a)Note that F2is a quasi-elementary quotient Hopf alge- bra of A, so HF2is included as one of the detecting spectra in Theorem 3.1* *.6 (as compared to [HSb ], when HF2is included for parts (a) and (c), but not for (b)). (b)Fix a quotient Hopf algebra A0of A, and let D0be the quotient of A0induced by A i D. One can generalize Theorem 3.1.5 in an obvious sort of way: if Y is a finite spectrum and f :Y -! HA0^ Y is a "self-map," then f 2[Y; HA0^ Y ]**is nilpotent , jD0^ f 2 [Y; HD0^ Y ]**is nilpotent , jE0^ f 2 [Y; HE0^ Y ]**is nilpotent forEall0: (Here E0ranges over all quasi-elementary quotients of A0.) There are simil* *ar versions of the ring spectrum and smash-nilpotence results. The proofs are straightforward generalizations of the ones below, so we omit them. 3.2. NILPOTENCE AND F-ISOMORPHISM VIA THE HOPF ALGEBRA D 51 (c)The quotient Hopf algebra D is "best possible," in the sense that if B is quotient of A which does not map onto D, then there are non-nilpotent elements in ss**S0 which are in the kernel of ss**S0 -!HB**. One can see this from Theorem 3.1.2. (d)Theorem 3.1.5 identifies a single ring spectrum HD which detects nilpotenc* *e, but we do not know its coefficient ring completely. See Propositions 3.3.4 and 4.4.1 for partial information. On the other hand, we have computed the coefficient rings of HE for E quasi-elementary in Proposition 2.1.9_they are polynomial rings. (e)We also mention that one does not need to use all of the quasi-elementary quotient Hopf algebras of A to detect nilpotence; for instance, one can use only the maximal quasi-elementary quotients, or only the finite-dimensional ones. (f)In fact, one can see from the proof of Theorem 3.1.6 that one only needs * *the following spectra to detect nilpotence: j+1 {h-1m+1;jH(E(m)=(2m+1)) | m 0; 0 j m}: 3.2. Nilpotence and F-isomorphism via the Hopf algebra D In this section we show that the spectrum HD detects a lot of information: we prove that it detects ss**S0 modulo nilpotent elements (Theorem 3.1.2), and tha* *t it detects non-nilpotent elements of ss**R for any ring spectrum R (Theorem 3.1.5). The Hopf algebra D is a conormal quotient of A, by Theorem 2.1.1. So there is a Hopf algebra extension A 2D Fp-! A -!D; and a spectral sequence (as in Section 1.5) with Es;t2= ExtsA2DFp(Fp; ExttD(Fp;)Fp)) Exts+tA(Fp; Fp): The restriction (Hurewicz) map factors through the edge homomorphism ExttA(Fp; Fp) -!E0;t2: This gives the factorization of the Hurewicz map h: ss**S0 -!HD**as advertised in Theorem 3.1.2: ': ss**S0 -!HDA2DF2**: It remains to prove Theorem 3.1.5_the spectrum HD detects nilpotence_and to verify conditions (i) and (ii) of Definition 3.1.1(b)_every element of ker' * *is nilpotent,mand for every element y 2 HDA2DFp**, there is an integer m so that y2 2 im'. Note that condition (i) follows from Theorem 3.1.5(a) with R = S0. Also, the proof of Theorem 3.1.5 and the verification of (ii) are quite similar, so first* * we lay the groundwork for both. For each integer n 1, we let n D(n) = A=(21; 42; : :;:2n): We let D(0) = A. See Figure 3.2.B. Each D(n) is a conormal quotient Hopf algebra of A, and we have a diagram of Hopf algebra surjections: A = D(0) i D(1) i D(2) i : ::: 52 3. QUILLEN STRATIFICATION AND NILPOTENCE | | | | | | | n|______| n-|1 | | | _________________________| p = 2 Figure 3.2.B. Profile function for D(n). As in Figure 2.1.B, the diagonal line is an abbreviation for a staircase shape. D is the colimit of this diagram. First we discuss how to lift information from HD**to HD(n)**for some n. We have the following lemma. Lemma 3.2.1. (a)We have HD**= lim-!HD(n)**. (b)We have HDA2DFp**= lim-!(HD(n)A2D(n)Fp**). Proof. Since homotopy commutes with direct limits, part (a) is clear. Part (b): The coaction of A on HD(n)**is defined in Remark 1.3.8. Note that each restriction map HD(n)**-! HD(n + 1)**is an A-comodule map (and in fact, a map of comodule algebras.) We take injective resolutions of these comodules, and apply ss0;*(-) (i.e., Hom *A(Fp; -)); since homotopy commutes with colimits, we have Hom A(Fp; HD**) = HomA (Fp; lim-!HD(n)**) = lim-!HomA(Fp; HD(n)**): Now, Hom A(Fp; HD(n)**) = HD(n)A**= HD(n)A2D(n)Fp**, by conormality, so_we have the desired result. |__| Now we discuss how to take information about HD(n)**and get information about HD(n - 1)**. (We want to know about ss**S0 = HD(0)**, so we will even- tually want to use downward induction on n.) Not only is each D(n) a conormal quotient of A, it is also a conormal quotie* *nt of D(n - 1). The Hopf algebra kernel of the quotient map is easy to identify: we have an extension of Hopf algebras n F2[2n] -!D(n - 1) -!D(n); where 2nnis primitive in the kernel. So given any D(n - 1)-comodule M, there is a change-of-rings spectral sequence (see (1.5.2)) with Es;t;u2(M) = Exts;uF2[2nn](F2; Extt;*D(n)(F2;)M)) Exts+t;uD(n-1)(F2; M): By Proposition 1.5.3, in the category Stable(D(n-1)) of cochain complexes of in* *jec- tive left D(n-1)-comodules, this spectral sequence is the same (up to regrading* *) as the HD(n)-based Adams spectral sequence. Because of this, for parts of the proof we will work in the category Stable(D(n-1)). We also use the grading on the spe* *c- tral sequence as given here; we do not use the Adams spectral sequence grading * *from Section 1.5. Throughout, we abuse notation somewhat, writing Ext**D(n)(F2; X) f* *or ss**(HD ^ X). 3.2. NILPOTENCE AND F-ISOMORPHISM VIA THE HOPF ALGEBRA D 53 We need to establish an important property of this spectral sequence, that it has a nice vanishing plane at some Er-term. We write Es;t;ur(X) for the spectral sequence converging to Exts+t;uD(n-1)(F2; X). Proposition 3.2.2.Fix a connective spectrum X and an integer m. For some r and some c, we have Es;t;ur(X) = 0 when ms + t - u > c. We prove this using Theorem 1.5.5, which says that vanishing planes in Adams spectral sequences are generic (Definition 1.4.7). (And again, this is an Adams spectral sequence, as long as we work in the category Stable(D(n - 1)).) We fix a connective spectrum X. Lemma 3.2.3.For each integer m, there is a finite D(n - 1)-comodule W so that Es;t;u2(W ^ X) = 0 when ms + t - u > c, where c depends only on the connectivity of X. (In Adams spectral sequence grading, this vanishing plane is of the form Ep;q;v2(W ^ X) = 0 when mp - q > 0, i.e., when p _-1__m(-p1+ v) + __1__m(-q1+ v): In particular, the coefficients _-1_m-1and __1_m-1) satisfies the hypotheses of* * Theo- rem 1.5.5.) Proof. Choose an integer j n so that |2jn| = 2j(2n - 1) > m, and let W = Wj= F2[2nn]=(2jn), with the apparent D(n - 1) 2D(n)Fp-comodule structure. Then Wj is a trivial D(n)-comodule (by definition), and as coalgebras we have n 2j F2[2n] ~=Wj F2[n ]: Hence the E2-term of the spectral sequence for Wj^ X looks like Es;t;u2~=Exts;uF2[2nn](F2; Extt;*D(n)(F2; Wj^)X ) ~=Exts;uF(F n; W Extt;*(F ; X)) 2[2n]2 j D(n) 2 ~=Exts;u (F ; Extt;*(F ; X)): F2[2jn]2 D(n) 2 The Hopf algebra F2[2jn] is |2jn|-connected, so if L is a comodule which is zero below degree t, then Exts;uF2[2j(F2; L) = 0 n ] when u < |2jn|s + t. Now note that Extt;*D(n)(F2; X) is zero below degree t + c* * for some c dependent only on the connectivity of X. |___| Next we need to show that X is in thick(Wj^ X). We start with the following lemma, which describes how to build Wj out of F2in a nice way. 54 3. QUILLEN STRATIFICATION AND NILPOTENCE Lemma 3.2.4.For j n, let Wj = F2[2nn]=(2jn). Then there is a short exact sequence of D(n - 1) 2D(n)F2-comodules 2j| 0 -!Wj-! Wj+1-! |n Wj-! 0: The connecting homomorphism in Ext**D(n-1)(F2; -) is multiplication by j ** hnj= [2n] 2 ExtD(n-1)(F2; F2): Replacing Wj by its injective resolution gives a cofibration sequence 2j| hnj 1;|nWj- -!Wj-! Wj+1-! Wj: (We are abusing notation a bit here, by writing Wj for both the module and its injective resolution. We will continute this practice for the remainder of * *this section.) Proof. This follows from Lemma 1.3.10. |___| Suppose that j n; then Theorem B.2.1(a) tells us that the element hnj 2 HD(n - 1)**is nilpotent. Hence we have the following: Lemma 3.2.5.If j n, then X 2 thick(Wj^ X). Proof. We show by downward induction on i that for j i n, Wi^ X is in thick(Wj^ X); the lemma is proved when i = n, since Wn = S0. The induction starts (trivially) with i = j. Suppose that i < j. By the cofibration sequence in Lemma 3.2.4, together with the nilpotence of hni,_we_see that Wi^ X 2 thick(Wi+1^ X), and hence in thick(Wj^ X) by induction. |__| Proof of Proposition 3.2.2.This follows immediately from Lemma 3.2.3,_ Lemma 3.2.5, and Theorem 1.5.5. |__| 3.2.1. Nilpotence: Proof of Theorem 3.1.5. Now we prove Theorem 3.1.5, and hence verify condition (i) of Definition 3.1.1(b). Proof of Theorem 3.1.5.The basic idea of the proof is, of course, based on that of the nilpotence theorem in [DHS88 ]. As in that proof, one can reduce to the ring spectrum case_part (a)_in which the ring is connective. So we let R be a connective ring spectrum, we fix ff 2 ss**R and assume that HD**ff is nilpote* *nt. By raising ff to a power, we may assume that HD**ff = 0. We want to show that ff is nilpotent in ss**R = HD(0)**R. By Lemma 3.2.1, since HD**y = 0, then we must have HD(n)**y = 0 for some n. We want to show that if HD(n)**y = 0, then HD(n - 1)**yj = 0 for some j; the result will follow by downward induction on n. Consider the change-of-rings spectral sequence Es;t;u2(R) = Exts;uF2[2nn](F2; Extt;*D(n)(F2;)R)) Exts+t;uD(n-1)(F2; R): Write z for HD(n - 1)**y. Since HD(n)**y = 0, then z must be represented by a class "z2 Ep;q;v2with p > 0. Choose an integer m so that mp + q - v > 0. Then for any c, we can find a j so that mpj + qj - vj > c: 3.2. NILPOTENCE AND F-ISOMORPHISM VIA THE HOPF ALGEBRA D 55 By Proposition 3.2.2 (with X = R), for some r and c we have Es;t;ur= 0 when ms+t-u > c. As noted above, we can choose j so that "zjlies above this vanishing plane, so at the Er-term for which we have the vanishing plane, "zjmust be zero. Modulo terms of higher filtration, "zjis zero at E1 , but the higher filtration* * pieces are also above the vanishing plane, and therefore zero. So zj = 0 in the_abutme* *nt, HD(n - 1)**, which is what we wanted to show. |__| 3.2.2. F-isomorphism: Proof of Theorem 3.1.2. We need to show that the map ': ss**S0 -!HDA2DF2** is an F-isomorphism. By Theorem 3.1.5, it is a monomorphism mod nilpotents, so we have to show that it is an epimorphism mod nilpotents; in other words, we ha* *ve to verify condition (ii) of Definition 3.1.1(b). Verification of condition (ii).Fix y 2 HDA2DF2i;j. We show that there is an integer m so that y2m 2 im'. By Lemma 3.2.1(b), there is an n so that y lifts to HD(n)A2D(n)F2**. (Alternatively, one can use Lemma 3.2.1(a) to lift * *y to HD(n)**for some n, and then use Lemma 3.2.6 to show that some power of that lift is invariant.) Now we show that some power of y lifts to HD(n - 1)A2D(n-1)* *F2**; since D(0) = A, then downward induction on n will finish the proof. Since y is invariant under the A 2D(n)F2-coaction, then it is also invariant* * under the coaction of D(n - 1) 2D(n)F2(since the latter is a quotient Hopf algebra of* * the former). So y represents a class at the E2-term of the change-of-rings spectral sequence Es;t;u2(F2) = Exts;uF2[2nn](F2; Extt;*D(n)(F2;)F2)) Exts+t;uD(n-1)(F2; F* *2): Also, by assumption, y lies in the (t; u)-plane, say y 2 E0;q;v2. Choose m lar* *ge enough so that m + q - v - 1 is positive. By Proposition 3.2.2 (with X = S0), we know that for some c and r, we have Es;t;ur= 0 for c < ms + t - u (hence the sa* *me as true for Es;t;ur0, for all r0 r). For each i 0, Proposition 1.5.4 tells us that the possible differentials on* * y2i are i j+1;2iq-j;2iv dj+1(y2 ) 2 Ej+1 ; for j 2i. Choose i so that 2i> max(r - 1; ___c_-_m___m)+;q - v - 1 and fix j 2i. Then we have a vanishing plane at the E2i+1-term (and hence at t* *he Ej+1-term); we claim that the element dj+1(y2i) lies above the vanishing plane,* * and so is zero. We just have to verify the inequality specified by the vanishing pl* *ane: m(j + 1) + 2iq - j -=2iv(m - 1)j + m + 2i(q - v) (m - 1)2i+ m + 2i(q - v) = 2i(m + q - v - 1) + m > ___c_-_m___m(+mq+-qv--v1- 1) + m = c: 56 3. QUILLEN STRATIFICATION AND NILPOTENCE Hence y2iis a permanent cycle. For degree reasons, it cannot be a boundary; hen* *ce it gives a nonzero element of E1 , and hence a nonzero element of HD(n - 1)**. It only remains to show that y2i, or at least some power y2i+j, is invariant* * under the A-coaction. Let ae: HD(n - 1)**-!HD(n)**denote the Hurewicz (restriction) map. This map detects nilpotence: if x 2 kerae, then x is nilpotent. (This foll* *ows from Remark 3.1.7(b), for instance; alternatively, this is the main inductive s* *tep in proving Theorem 3.1.5.) Hence by Lemma 3.2.6 below, y2i+jis invariant for some j. This completes the verification of condition (ii) of Definition 3.1.1(b),_an* *d hence the proof of Theorem 3.1.2. |__| We have used the following. Lemma 3.2.6.Suppose that R and S are commutative A-comodule algebras, with an A-linear map ae: R -!S that detects nilpotence: every x 2 kerae isnnilp* *otent. Given z 2 R so that ae(z) 2 S is invariant under the A-coaction, then z2 is al* *so invariant, for some n. Proof. Since ae(z) is invariant, then the coaction on z is of the form X z 7-! 1 z + ai xi; i where each xiis in kerae, and hence is nilpotent. Since R is an A-comodule alge* *bra, then we see that n 2n X 2n 2n z2 7-! 1 z + ai xi : i This is a finite sum (since these are comodules), so for n sufficiently_large, * *z2n is invariant. |__| 3.3.Nilpotence and F-isomorphism via quasi-elementary quotients In this section we use our F-isomorphism and nilpotence theorems for HD_ Theorems 3.1.2 and 3.1.5_to prove analogous theorems for the quasi-elementary quotients of A_Theorems 3.1.3 and 3.1.6. 3.3.1. Nilpotence: Proof of Theorem 3.1.6. We start with Theorem 3.1.6, because we use it in the proof of Theorem 3.1.3. We need a few preliminary resu* *lts. Suppose we have a map f :S0 -!X. We define X(1) to be the sequential colimit of the following diagram: S0 f-!X f^1--!X ^ X f^1^1----!X ^ X ^ X -!: :;: and we let f(1): S0 -!X(1) be the obvious map. We recall the following from [HSb , Lemma 2.3]. Lemma 3.3.1.Given a map f :S0 -!X and a ring spectrum E with unit map j :S0 -!E, the following are equivalent. (a)E ^ X(1)= 0. (b)j ^ f(1): S0 -!E ^ X(1) is zero. (c)j ^ f(n):S0 -!E ^ X(n)is zero for n 0. (d)1E ^ f(n):E = E ^ S0 -!E ^ X(n)is zero for n 0. 3.3. NILPOTENCE VIA QUASI-ELEMENTARY QUOTIENTS 57 | | | ___ | ___ | ___| | ___| | | | ___| | ___| | | | | ___| | :.|: : | | ..| | ___| | .:.: |: | ___| | : :.: | | | .. | | ___| | __________|. | | | | | :.|:.: | .:.|: : |:.:|:D2 |:.: |:D2;6 | | | | _________________________|| _________________________|| Figure 3.3.C. Profile functions for Dr and Dr;q. For integers q > r 0, we define the following quotient Hopf algebras of D: Dr = D=(1; : :;:r); r+1 2r+1 Dr;q= Dr=(2r+2; : :;:q ): See Figure 3.3.C. Recall from Corollary 2.1.8 that the maximal quasi-elementary quotients of A are called E(m), m 0. Lemma 3.3.2.We have lim-!rHDr = HF2and lim-!qHDr;q= HE(r). Proof. We leave this as an exercise. |___| Lemma 3.3.3.Let i, j, q, r, and R be integers. (a)Suppose that q > r i 0 and q - r > j 0, and consider the Hopf algebra B = Dr;q=(2i+1r+1; 2r+2+jq+1). By Lemma 2.1.2, there are Hopf alge* *bra extensions i E[2r+1] -!B -!C1; r+1+j E[2q+1 ] -!B -!C2; leading to elements hr+1;iand hq+1;r+1+jin HB**. Then hr+1;ihq+1;r+1+j is nilpotent. (b)Whenever q > r and r i 0, there is a Bousfield equivalence i+1 -1 2i+1 = : (c)Whenever q > R, we have a Bousfield class decomposition R_ _r i+1 = _ r=0i=0 R_ _r i+1 = _ : r=0i=0 Proof. For (a), the nilpotence of hr+1;ihq+1;r+jis due to Lin [Lin]; we reca* *ll the statement of this result as Theorem B.2.1(b). (The nilpotence of this produ* *ct is also the content of [Wil81 , Theorem 6.4], as well as being essentially equi* *valent to the classification of quasi-elementary quotients of A in Propositions 2.1.7 * *and 2.1.12.) 58 3. QUILLEN STRATIFICATION AND NILPOTENCE Part (b) follows from part (a) and Corollary 1.6.2, by induction: one can get from Dr=(2i+1r+1) to Dr;q=(2i+1r+1) by dividing out by one 2stat a time, where * *s r+1. In other words, one has a sequence of extensions of the form s E[2t] -!B -!C; and hence cofibrations of the form HB hts--!HB -!HC: One inverts hr+1;iin each term; then by the nilpotence of hr+1;ihts, one gets an equivalence of Bousfield classes = : Part (c) is similar to part (b). |___| Proof of Theorem 3.1.6.We imitate the proof of [HSb , Theorem 3]. As in that proof, one can reduce to the smash-nilpotence case, and using Spanier- Whitehead duality, one can reduce to the case where F = S0. Suppose we have a map f :S0 -!X so that jE ^ f = 0 for all quasi-elementary E. We want to show that j ^ f(n):S0 -! HD ^ X(n)is zero for some n; then Theorem 3.1.5(c) will tell us that f is smash-nilpotent. By Lemma 3.3.1, this map is zero if and only* * if HD ^ X(1)= 0. We use a Bousfield class argument to show this. By assumption, HE^X(1)= 0 for all quasi-elementary E. By Lemma 3.3.3(c), we must show that HDR+1 ^ X(1)= 0; for some R, and then that i+1 (1) h-1r+1;iH(Dr;q=(2r+1)) ^ X = 0; for all r R, i r, and q 0. First we show that HDR+1 ^ X(1) = 0. By Lemma 3.3.1, it is equivalent to show that j ^ f(1): S0 -!HDR+1 ^ X(1) is zero. Let R go to infinity; then by Lemma 3.3.2, the map S0 -!lim-!RHDR+1 ^ X(1)= HF2^ X(1) is null. Since homotopy commutes with direct limits, then for some R, the map S0 -!HDR+1 ^ X(1) is null. One uses the same argument to show that i+1 (1) h-1r+1;iH(Dr;q=(2r+1)) ^ X = 0; but using the second equality of Lemma 3.3.2, rather than the first. |* *___| 3.3.2. F-isomorphism: Proof of Theorem 3.1.3. Now we work on Theo- rem 3.1.3: the map i jA ss**S0 -! lim-QHE** is_an F-isomorphism. First we construct the coaction of A on lim-QHE**. We let Q denote the full subcategory of Q consisting of the conormal quasi-elementary quotient Hopf algebras_of_A. Since the maximal quasi-elementary quotients are conormal, we see that Q is final in Q; hence we have lim-QHE**= lim-_QHE**: 3.3. NILPOTENCE VIA QUASI-ELEMENTARY QUOTIENTS 59 On the right we have an inverse limit of comodules over A; we give this the ind* *uced A-comodule structure. So we have (since taking invariants is an inverse limit) i jA i Aj lim-QHE** = lim-_QHE** A = lim-_QHE** i j = lim-_QHEA==E**: To finish the proof of Theorem 3.1.3, we show that we can compute HD**up to F-isomorphism in terms of the coefficient rings HE**for E quasi-elementary. Proposition 3.3.4.The natural map HD**-! lim-QHE**; is an F-isomorphism, as is the induced map i jA HDA**-! lim-QHE** : See Proposition 4.4.1 for an explicit computation of the ring lim-QHE**. Proof of Theorem 3.1.3.This follows immediately from Theorem 3.1.2_and Proposition 3.3.4. |__| Proof of Proposition 3.3.4.We need to show that the restriction map ae: HD**-! lim-HE** E2Q is an F-isomorphism, so we need to show two things: every element in the kernel of f is nilpotent, and we can lift some 2nth power of any element in the range * *of f. By the nilpotence theorem 3.1.6 (or more precisely, by the generalization in Remark 3.1.7(b)), we know that an element z 2 HD**is nilpotent if and only if zn 2 kerae for some n. The second statement follows, almost directly, from a result of Hopkins and Smith [HSb , Theorem 4.12], stated as Theorem 3.3.5 below, combined with the fact that the Hopf algebra D is a direct limit of finite-dimensional Hopf algeb* *ras. We explain. For each integer r 1, we let B(r) be the following quotient Hopf algebra of D: B(r) = D=(1; 2; : :;:r): Then each B(r) is conormal in D, the kernel K(r) = D 2B(r)F2is finite-dimension* *al, and we have D = lim-!rK(r). Note that this colimit stabilizes in any given degr* *ee. Given a quasi-elementary quotient E of D, we define the quotient F(r) of E to be the pushout of B(r) - D -!E, and we let E(r) be the Hopf algebra kernel of E -! F(r). In other words, we have the following diagram of Hopf algebra extensions: K(r) ----! D ----! B(r) ?? ? ? y ?y ?y E(r) ----! E ----! F(r): 60 3. QUILLEN STRATIFICATION AND NILPOTENCE Since E = lim-!rE(r), then given an element y 2 HE**, for r sufficiently large, y is in the image of the inflation map Ext**E(r)(F2; F2) -! Ext**E(F2; F2). No* *w, K(r) -!E(r) is a quotient map of finite-dimensional graded connected commuta- tive Hopf algebras; let resK(r);E(r)denote the induced map on Ext. Then given y 2 Ext**E(r)(F2; F2), for some n we have ypn 2 im(resK(r);E(r)), by Theorem 3.* *3.5. The following commutative diagram finishes the proof of the first statement: Ext**K(r)(F2;-F2)---!Ext**D(F2; F2) ?? ? y ?y Ext**E(r)(F2;-F2)---!Ext**E(F2; F2): Now we want to show that the natural map HDA2DF2**-!lim-_QHEA2EF2** is an F-isomorphism. We treat the A 2D F2-coaction on HD** as an A-coaction with trivial D-coaction (and similarly for A 2EF2coacting on HE**). So we want to compare HDA**with lim-HEA**. We write f and "ffor the maps f :HD**-! lim-_QHE**; "f:HDA**-!lim__HEA : - Q ** The maximal quasi-elementary quotients are conormal; hence the conormal quasi- elementary quotients are final in the inverse system of all quasi-elementary qu* *o- tients; hence f is an F-isomorphism. So it is clear that if x 2 HDA**is in the * *kernel of "f, then x is nilpotent. Furthermore, given y 2 lim-HEA**, we know that y is* * in the image of f; hence by Lemma 3.2.6, some 2nth power of y must be_in_the image of "f. |__| We have used the following theorem. This first appeared in [HSb ], and is a generalization of results in [Wil81 ]. In this theorem, Ext is taken in the cat* *egory of modules, rather than comodules; we have actually used the theorem which is dual to this one. Theorem 3.3.5 (Theorem 4.12 in [HSbS]).uppose that is an inclusion of finite-dimensional graded connected cocommutative Hopf algebras over a field* *mk of characteristic p > 0. For any 2 Ext**(k; k), there is a number m so that p is in the image of the restriction map Ext**(k; k) -!Ext**(k; k). 3.4.Further discussion: nilpotence at odd primes There are two main obstructions to proving the above results at odd primes. The first is clear: we do not have a classification of the quasi-elementary quo* *tients of A; hence we cannot prove the quasi-elementary versions of the theorems. (Equ* *iv- alently, we do not have an odd primary analogue of Theorem B.2.1(b).) The second is perhaps more important, since it affects the HD versions of the theorems, and more surprising, because the corresponding result at the prime 2 seems rather s* *tan- dard. This is the propertyjthat allows one to prove Lemma 3.2.5: at the prime 2, the element hnj= [2n] 2 HD(n)**is nilpotent, if j n (see Theorem B.2.1). Recall from Notation 1.3.9 and Lemma 1.3.10 that primitives y in a Hopf al- gebra B give rise to classes [y] 2 Ext1;|y|B(k; k). When p is odd, even-dimensi* *onal primitives y in B give rise to classes fifP0[y] in Ext2;p|y|B(k; k). 3.5. FURTHER DISCUSSION: MISCELLANY 61 Conjecture 3.4.1.Fixsan odd prime p. Fix a quotient Hopf algebrasB of A, and suppose that pt is primitive in B. If s t, then bts= fifP0[pt] is nilpotent in HB**. As at the prime 2, we define D as follows: 2 p3 D = A=(p1; p2; 3 ; : :):: One can see from the proofs when p = 2 that the odd prime versions of Theo- rems 3.1.2 and 3.1.5 would follow from this conjecture. (One has to make a slig* *ht change in Lemma 3.2.4_as in the proof of Lemma 1.3.10, there are two related short exact sequences of comodules, leading to a cofibration in which the conne* *ct- ing homomorphism is a map of homological degree 2. Other than that, everything goes through as written.) We discuss the status of Conjecture 3.4.1 in Appendix B.3. 3.5. Further discussion: miscellany The nilpotence theorem [DHS88 ] in stable homotopy theory has far-reaching implications for the global structure of the stable homotopy category_see [Hop8* *7 ] and [Rav92 ], for instance. We develop analogues of some of the structural resu* *lts in the next chapter, but there are still many gaps. We discuss those in Section* *s 4.6 and 4.9 below. If is a Hopf algebra, then the quasi-elementary quotients of give the right homology functors to consider for detecting nilpotence, essentially by the defi* *nition of "quasi-elementary." Indeed, if is commutative and finite-dimensional, it is* * not too hard to prove an analogue of Theorem 3.1.6, either directly (as in [Pal97]) or via Chouinard's theorem (as in [HPS97 , 9.6.10-9.6.11]). If, in addition, is connected, then Theorem 3.3.5 allows one to prove an analogue of Theorem 3.1.3_ see [Pal97]. Actually, by Theorem 3.3.5, some power of every element in the inv* *erse limit is invariant, so one gets an F-isomorphism Ext**(k; k) -!lim-QExt**E(k; k); where the inverse limit is over Q, the category of quasi-elementary quotients o* *f . Here are some related questions: one already has Quillen stratification for * *group algebras; can one prove it, as we do, with vanishing lines or planes? Can one u* *se our approach to Quillen stratification for the Steenrod algebra to study the cohomo* *logy of other Hopf algebras? Given a Hopf algebra , one should study the pullback of the quasi-elementary quotients of , so one might want to assume that those quotients are somewhat well-behaved. Even more generally, in an arbitrary stable homotopy category one could consider a subcategory of appropriately chosen ring spectra, and look at the inverse limit of that. Can one refine the description of ss**S0 as given by Theorem 3.1.2? In parti* *c- ular, can one describe the nilpotence height of elements in the kernel, or say * *which powers of classes in the codomain of ': ss**S0 -!HDA2DF2** lift to the domain? Since one has a version of Quillen stratification for the Steenrod algebra, * *one might look for analogues of other group-theoretic results. Chouinard's theorem [Cho76 ] is an example: a kG-module M is projective if and only if it is projec* *tive upon restriction to kE for every elementary abelian subgroup E of G. An analogue 62 3. QUILLEN STRATIFICATION AND NILPOTENCE in Stable(A) might be: for any spectrum X, X is in loc(A) if and only if HD ^ X* * is in loc(A) if and only if HE ^ X is in loc(A) for every quasi-elementary quotien* *t E of A. While one can prove things like this in Stable() for finite-dimensional H* *opf algebras _see [Pal97]_the infinite-dimensionality of A might cause a problem. One should be able to prove something like this when X is a finite spectrum, but one would like a version of Chouinard's theorem without any such restrictions. CHAPTER 4 Periodicity and other applications of the nilpotence theorems The nilpotence theorem in ordinary stable homotopy theory [DHS88 ] has a number of important consequences: the periodicity theorem and the thick subcat- egory theorem of [HSb ] are examples. In this chapter we study applications of * *our nilpotence and Quillen stratification theorems_Theorems 3.1.2, 3.1.3, 3.1.5, and 3.1.6. One of our main results of this chapter is a version of the periodicity theo* *rem: if R is a finite ring spectrum, then we produce a number of central non-nilpote* *nt elements in ss**R via the "variety of R over D," which is essentially the kerne* *l of HD**j :HD**-! HD**R. Equivalently, this gives families of central non-nilpotent self-maps of any finite spectrum X. This is our analogue of the periodicity the* *orem of Hopkins and Smith. We state this precisely in Section 4.1, and we prove it in Sections 4.2 and 4.3. Theorem 3.1.2 says that ss**S0 is F-isomorphic to the A-invariants in HD**; * *in Section 4.4 we discuss some examples of invariant elements in HD**. W In Section 4.5, we show that the objects that detect nilpotence_HD and HE_have strictly smaller Bousfield classes than that of the sphere. The role of D (as well as the action of A on D) has led us to a conjectured thick subcat- egory theorem, which we give in Section 4.6. We also discuss a few properties of "varieties" of spectra over D in this section. In Section 4.7 we construct finite spectra (analogues of generalized Toda V * *(n)'s) with vanishing lines of various slopes, using Theorem 2.4.3, and we examine some properties of these spectra. These will be used in studying chromatic phenomena in Chapter 5. We end the chapter with two sections of miscellany_one on slope supports of finite spectra, and a very brief section with a few additional ques* *tions and remarks. As in the previous chapter, we work at the prime 2 (unless otherwise stated). 4.1.The periodicity theorem We start this chapter by giving our version of the periodicity theorem. This* * is a weak analogue of Theorem 3.1.2, with nontrivial coefficients. The following definition was motivated by work in modular group represen- tation theory of Alperin, Benson, Carlson, and the rest of the group theoretic alphabet. Definition 4.1.1.Given a finite spectrum X, we define the ideal of X, I(X), to be the radical of i j kerHD**--^X--![X; HD ^ X]**: 63 64 4. APPLICATIONS OF THE NILPOTENCE THEOREMS Since X is finite, this is the same as the radical of the annihilator ideal * *in HD** of HD**X via the action by composition (see Proposition 4.6.2). The ideal I(X) * *is also invariant under the A-coaction, so that the A-coaction on HD**induces one on HD**=I(X). Definition 4.1.2.Given an element y 2 Ext**D(F2; F2) = HD**and an object X in Stable(B), a map z :X -! X is a y-map if HD**z :HD**X -! HD**X is multiplication by yn for some n. Theorem 4.1.3 (Periodicity theorem).Let X be a finite spectrum. For every i jA y 2 HD**=I(X) ; X has a y-map which is central in the ring [X; X]**. (There is an equivalent statement involving elements in the homotopy of a fi* *nite ring spectrum R, which we give as Theorem 4.2.2 below.) We conjecture that (HD**=I(X))Ais F-isomorphic to the center of [X; X]**. See Section 4.6 for this and related ideas. As an application, we have Corollary 4.1.5 below, which first appeared as [Pal96b, Theorem 4.1]. We need a bit of notation to state it. Definition 4.1.4.Let Slopesdenote the set of slopes of A (Notation 2.2.4). Given a spectrum X, we define its slope support to be the set {n | Z(n)**X 6= 0} Slopes: Since we are working at the prime 2, we have a bijection Slopes! {(t; s) | t > s 0}; s |2t|! (t; s): Let Slopes0denote the right-hand side of this bijection. We say that a subset T* * of Slopes0is admissible if T satisfies the following conditions: (t; s) 2 T ) (t + 1; s) 2 T; (t; s) 2 T ) (t + 1; s - 1) 2 T; card(Slopes0\ T) < 1: (We call such sets "admissible" because those are the possible slope supports of finite spectra_see Section 4.8 and [Pal96b].) Recall from Notations 2.2.4 and 2.4.1 that for each slope n, there is a spec* *trum Z(n) and a non-nilpotent element un 2 Z(n)**. There is a ring map HD** -! Z(n)**, and we show in Section 4.4 that some power of un lifts to HD**. Hence we can consider the existence of un-maps,a la Definition 4.1.2. Corollary 4.1.5.Let X be a finite spectrum and let T Slopesbe the slope support of X. If n 2 T and T \ {n} is admissible (viewed as a subset of Slopes0* *), then there is a (non-nilpotent) un-map in [X; X]**. Proof. One can see that I(X) contains {ht;s| (t; s) 2 T Slopes0}, and_ hence un is invariant in HD**=I(X). |__| 4.2. PROPERTIES OF y-MAPS 65 4.2.Properties of y-maps In this section we lay the groundwork for proving Theorem 4.1.3. In particul* *ar, we study analogues of vn-maps [Hop87 , HSb ]. In ordinary stable homotopy the- ory, vn-maps are defined via the Morava K-theories_these are field spectra, and hence have various convenient properties, such as K"unneth isomorphisms. Our an* *a- logues are defined via HD, and hence are not quite as easy to work with. Noneth* *e- less, our versions of vn-maps, called "y-maps," share many of the same properti* *es as vn-maps in ordinary stable homotopy theory; in particular, we show in this sect* *ion that the property of having a y-map is generic in our setting. Fix a quotient Hopf algebra B of A which maps onto D. We work in the category Stable(B). We start by expanding Definition 4.1.2 a bit: Definition 4.2.1.Fix y 2 Ext**D(Fp; Fp) = HD**, and let X be an object in Stable(B). A map z :X -! X is a y-map if HD**z :HD**X -! HD**X is multiplication by yn for some n. Similarly, if R is a ring object in Stable(B),* * then an element ff 2 ss**R is a y-element if for some n, we have HD**ff = yn as maps HD**-! HD**R. Note that the set of y-maps from X to itself is in bijection with the set of y-elements in the ring spectrum X ^ DX, by Spanier-Whitehead duality. So the following is equivalent to Theorem 4.1.3. Theorem 4.2.2 (Periodicity theorem for ringpspectra).Let_R be a finite ring spectrum with unit map j :S0 -!R. Let I = kerHD**j. For every i jA y 2 HD**=I ; there is a y-element which is central in ss**R. The main tool for proving Theorems 4.1.3 and 4.2.2 is the following. See Def* *i- nition 1.4.7 for the definition of "thick subcategory." Theorem 4.2.3.Suppose that B is a quotient Hopf algebra of A with A i B i D. Fix y 2 HD**. The full subcategory C consisting of finite objects of Stable(B) having a y-map is thick. In other words, the property of having a y-map is generic. The proof is a simple modification of the proof in [HSb ] that having a vn-m* *ap is a generic property. We devote most of this section to the details. We start * *with a variant of the notion of y-map, and a general lemma. Suppose that we have B i D. Since a change-of-coalgebras isomorphism (as in Lemma 1.3.4) gives Ext**B(M; (B 2D Fp) M) ~=Ext**D(M; M); it is sometimes useful to consider [X; HD ^ X]**. Note that the ring HD** = [S0; HD]**acts on this, via the smash product. We have the following. Definition 4.2.4.Fix y = HD**, and let X be an object in Stable(B). Write j :S0 -!HD for the unit map. A map z :X -! X is a strong y-map if j ^ z = yn ^ 1 2 [X; HD ^ X]**for some n. For instance, the map produced in Theorem 2.4.3 is a strong y-map (where y = un), and hence a y-map by the following lemma. 66 4. APPLICATIONS OF THE NILPOTENCE THEOREMS Lemma 4.2.5.Let X be an object in Stable(B). If z :X -! X is a strong y-map, then it is a y-map. Proof. Since j is the unit map of the ring spectrum HD, then the following composite equals 1 ^ z: HD ^ X 1^j^z----!HD ^ HD ^ X ^1--!HD ^ X: By assumption, 1 ^ j ^ z = 1 ^ yn ^ 1, so this composite also equals yn ^ 1. We compute the induced map on ss**by n^1 S0 -!HD ^ X y---!HD ^ X: Hence z is a y-map. |___| We move on to the proof of Theorem 4.2.3. Here is an easy lemma. Lemma 4.2.6.Let p be a prime. Fix elements y1 andny2nof an Fp-algebra. If y1 and y2 commute and y1- y2 is nilpotent, then yp1= yp2 for some n 0. For the remainder of the section, we consider a fixed element y of HD**. We work at the prime 2. Lemma 4.2.7.Let R be a ring spectrum in Stable(B), and fix y-elements ff; fi* * 2 ss**R. If ff and fi commute, then there exist positive integers i and j so that* * ffi= fij. Proof. We may assume that HD**(ff - fi) = 0 by raising ff and fi to suitable powers; hence ff-fi is nilpotent. Since ff and fi commute, then Lemma_4.2.6 fin* *ishes the proof. |__| Lemma 4.2.8.Let R be a finite ring object in Stable(B), and fix a y-element ff 2 ss**R. For some i > 0, the element ffiis central in ss**R. Proof. Let `(ff); r(ff) 2 End(ss**R) denote left and right multiplication by* * ff, respectively. More precisely, `(ff) is induced by the following self-map of R: R ff^1--!R ^ R -!R; and similarly for r(ff). Since HD**ff is central in HD**R, then `(ff) - r(ff) m* *aps to zero in End(HD**R), and so is nilpotent by Theorem 3.1.5. By Lemma 4.2.6,_ we conclude that ff is central in ss**R. |__| Corollary 4.2.9.Let R be a finite ring spectrum in Stable(B). For any y- elements ff; fi 2 ss**R, there exist positive integers i and j so that ffi= fij. Corollary 4.2.10.Let X be a finite spectrum in Stable(B), and let f and g be two y-maps of X. Then fi = gj for some positive integers i and j. Corollary 4.2.11.Suppose that X1 and X2 have y-maps y1 and y2. Then there are positive integers i and j so that for every Z and every f : Z ^ X1 -!* *X2, the following diagram commutes: Z ^ X1 ----! X2 ? f ? 1^yi1?y ?yyj2 Z ^ X1 ----! X2 f 4.3. THE PROOF OF THE PERIODICITY THEOREM 67 Proof. DX1 ^ X2 has two y-maps: Dy1^ 1 and 1 ^ y2. Now we apply Corol-_ lary 4.2.10 and Spanier-Whitehead duality. |__| Proof of Theorem 4.2.3.If Y has a (central) y-map gY and if X is a retract of Y , then the induced self-map of X X -!Y -gY-!Y -! X is easily seen to be a y-map. Suppose that X1 -! X2 -! X3 is a cofibration, and that X1 and X2 have y-maps y1 and y2, respectively. We may assume that HD**y1 and HD**y2 both are multiplication by yn. By Corollary 4.2.11, we can find a map y3 so that this diagram commutes: X1 ----! X2 ----! X3 ?? ? ? yy1 ?yy2 ?yy3 X1 ----! X2 ----! X3 We claim that some power of y3 is a y-map. So we compare HD**(y3) and yn in End(HD**X3). We have the following commutative diagram: HD ^ X2 ----! HD ^ X3 ----! HD ^ X1: ? ? ? 1^y2-yn^1?y ?y1^y3-yn^1 ?y1^y1-yn^1 HD ^ X2 ----! HD ^ X3 ----! HD ^ X1 Since y1 and y2 are y-maps, the left- and right-hand vertical maps induce zero * *on ss**, so by a simple diagram chase, one can see that ss**(1^y3-yn^1)2_= 0. Hence y23is a y-map. |__| 4.3.The proof of the periodicity theorem In this section we prove Theorem 4.1.3. Fix a finite object X in Stable(A). * *For each element y 2 (HD*=I(X))A, we want to show that X has a y-map (Defini- tion 4.1.2). The basic pattern of the proof is the same as that of Theorem 3.1.2: we indu* *c- tively work our way from D to A via the Hopf algebras D(n) (defined in Section * *3.2). As in Lemma 3.2.1, since X is finite, then [X; HD ^ X]** is the colimit of [X; HD(n)^X]**. So for some n sufficiently large, the element j^y 2 [X; HD^X]** lifts to a strong y-map in [X; HD(n) ^ X]**= [X; X]D(n)**. By Lemma 4.2.5, this gives a y-map in [X; X]D(n)**, and by Lemma 4.2.8, we may assume that this y-map is central. This starts the induction. Now, assume that X has a y-map when viewed as an object in Stable(D(n)). We want to show that X still has a y-map, but when viewed as an object in Stable(D(n - 1)). In the latter category, for any object Y we have the Adams spectral sequence based on HD(n): Exts;uD(n-1)2D(n)F2(F2; Extt;*D(n)(F2;)Y))Exts+t;uD(n-1)(F2; Y ): 68 4. APPLICATIONS OF THE NILPOTENCE THEOREMS In particular, if we let Wj be defined as in Lemmas 3.2.3 and 3.2.4, then we ha* *ve (4.3.1) Exts;uD(n-1)2D(n)F2(F2; Extt;*D(n)(F2; X ^ Wj^ DX ^)DWj) ) Exts+t;uD(n-1)(F2; X ^ Wj^ DX ^ DWj) = [X ^ Wj; X ^ Wj]D(n-1)**: Lemma 3.2.5 tells us that thick(S0) = thick(Wj) for any j n, so thick(X) = thick(X ^ Wj). Hence by Theorem 4.2.3, it suffices to show that X ^ Wj has a y-map for some j. The idea is that for j sufficiently large, the endomorphisms of X ^ Wj over D(n - 1) should be more or less the same as the endomorphisms of X over D(n). Since X has a y-map over D(n), then X ^ Wj should have one over D(n - 1), and hence X should have one, by genericity. The details are as follows. Since X is finite, then there is a number a so t* *hat Exts;uD(n)(F2; X ^ DX) is zero when u-a < s. The comodule Wjis nonzero between degrees 0 and 2j|n|, so Exts;uD(n)(F2; X ^ Wj^ DX ^ DWj) is zero when u - (a - 2j|n|) < s. By Lemmas 3.2.5 and 3.2.3, then, we see that * *the E2-term for the Adams spectral sequence (4.3.1)has the following vanishing plan* *e: Es;t;u2(X ^ Wj^ DX ^ DWj ) = 0 when 2j|n|s + t - u - (2j|n| - a) > 0: The inductive hypothesis tells us that we have a y-map in Ext**D(n)(X; X). We u* *se "yto denote this element, as well as its image in Ext**D(n)(X ^ Wj; X ^ Wj). No* *w, Ext**D(F2; F2)=I(X) -!Ext**D(n)(X ^ Wj; X ^ Wj) is a map of D(n-1) 2D(n)F2-comodules, so since y is assumed to be in the invari* *ants of Ext**D(F2; F2)=I(X), then "yis invariant under the D(n - 1) 2D(n)F2-coaction. Hence "yrepresents an element at the E2-term of the spectral sequence. We claim that, when j is large enough, "yis a permanent cycle. Suppose that "y2 Extp;qD(n)(X; X); then it gives a class in E0;p;q2. The rth differential on this class would lie in Er;p-r+1;qr. So we only have to check t* *hat for all r 2, this group is above the vanishing plane, and hence zero. We check our inequality: ? 2j|n|r + (p - r + 1) - q - (2j|n| - a) > 0: Whatever p, q, and a are, we can choose j large enough so that this holds for all r 2. Hence "yis a permanent cycle in the spectral sequence for X ^ Wj; it obviously cannot support a differential, so it survives to give a nonzero class* * at E1 . Since the resulting self-map of X over D(n - 1) restricts to the y-map "yover D* *(n), then one can check that it is a y-map over D(n - 1). This completes the inducti* *ve step, and with it, the proof of Theorem 4.1.3. Remark 4.3.2.We have used the nilpotence part of the "Quillen stratification" theorem 3.1.2 (i.e., we have used Theorem 3.1.5) in the proof of Theorem 4.2.3,* * and hence in the proof of the periodicity theorem 4.1.3. We have not used the other part of Theorem 3.1.2_that some power of any invariant element in HD**lifts to ss**S0. Indeed, this follows from the periodicity theorem, so it gives us an al* *ternate proof of Theorem 3.1.2. 4.4. COMPUTATION OF SOME INVARIANTS IN HD** 69 4.4.Computation of some invariants in HD** Theorem 3.1.3 gives an F-isomorphism ss**S0 -!(lim-QHE**)A: We compute lim-HE** in Proposition 4.4.1 below; in Proposition 4.4.4 we give a formula for the coaction of A on this inverse limit, and then we give a few exa* *mples of invariant elements. The maximal quasi-elementary quotients of A when p = 2 are the Hopf algebras E(m), m 0. We recall from Corollary 2.1.8 and Proposition 2.1.9 their definiti* *on and the computation of their coefficient rings: m+1 2m+1 2m+1 E(m) = A=(1; : :;:m ; 2m+1; m+2 ; m+3 ; : :):; HE(m)**= F2[hts| t m + 1; s m]: The bidegrees of the polynomial generators are given by |hts| = (1; |2st|). Sin* *ce it is easy to see the effects of the maps in Q on the coefficients, we immediately* * have the following. Proposition 4.4.1.There is an isomorphism ~= lim-HE**-! F2[hts| s < t] = (htshvu | u t); where |hts| = (1; |2st|). Proposition 3.3.4 gives us an F-isomorphism HD**-! lim-HE**, which we can compose with this isomorphism; hence, to compute HD** up to F-isomorphism, one does the following: oone takes the coefficient rings HE**, as E ranges over the maximal quasi- elementary quotients of A, otensors them together, oidentifies polynomial generators from different E's if they have the same name, and odivides out by the product of two polynomial generators if they do not come from the same E. We want to describe the coaction of A on lim-HE**. The coaction of A 2E(m)F2 on HE(m)** is determined by the coaction on the polynomial generators hts. Proposition 4.4.2.Fix m 0. Let O: A -! A denote the conjugation map of A, and for n 1 let in = O(n). Let i0 = 1. Under the coaction map HE(m)**-! (A 2E(m)F2) HE(m)**; we have m-sXt-jX s i+j+s hts7-! i2j2t-i-j hi;j+s: j=0i=m+1 (The ij part comes, essentially, from the right coaction of A on itself, whi* *le the t-i-jcomes from the left coaction.) Proof. (We assume that t > m + 1 and that s < m; the special cases when t = m + 1 or s = m are even easier to deal with.) First we find allnof the term* *s in the coaction htsof the form ahijfor a 2 A primitive, i.e., a = 21for some n. For 70 4. APPLICATIONS OF THE NILPOTENCE THEOREMS degree reasons, the only possible such terms are 2s1ht-1;s+1and 2s+t-11ht-1;s. By computations as in the proofs of Lemmas A.3 and A.5 of [Pal96b], we see that both of these terms do appear; that is, we can see that s 2s+t-1 hts7-! 1 hts+ i21 ht-1;s+1+ 1 ht-1;s+ other terms; where the "other terms" are of the form b hij, with b 2 A non-primitive. So the formula given in the proposition is "correct on the primitives"; once we ha* *ve verified that the formula is co-associative, we will have finished the proof. * *This verification is a straight-forward (although slightly messy) computation; it co* *uld be left to the diligent reader, but due to lack of space constraints,_we includ* *e it in Lemma 4.4.3 below. |__| Lemma 4.4.3.The formula m-sXt-jX s i+j+s hts7-! i2j2t-i-j hi;j+s: j=0i=m+1 defines a coassociative coaction of A on HE(m)**. Proof. We write for the coproduct on A, and we write for the coaction map of A on HE(m)**. We have to verify that ( 1) O = (1 ) O , so we compute both of these on hts. First, we have ( 1) O (hts): 0 1 m-sXXt-j s i+j+s ( 1) ( (hts)) = ( 1) @ i2j2t-i-j hi;j+sA j=0i=m+1 m-sXt-jX s i+j+s = (i2j)(2t-i-j) hi;j+s j=0i=m+1 m-sXt-jX Xj s s+n! t-i-jXi+j+s+q i+j+s! = i2n i2j-n 2t-i-j-q 2q hi;j+s j=0i=m+1 n=0 q=0 m-sXt-jXXj t-i-jXs i+j+s+q s+n i+j+s = i2n2t-i-j-q i2j-n2q hi;j+s: j=0i=m+1n=0 q=0 Note that we can read off the "coefficient" of ha;b+s: it is Xbt-a-bX s a+b+s+q s+n a+b+s i2n2t-a-b-q i2b-n2q : n=0 q=0 4.4. COMPUTATION OF SOME INVARIANTS IN HD** 71 On the other hand, for (1 ) O (hts) we have m-sX t-jX s i+j+s (1 )( (hts)) = (1 )( i2j2t-i-j hi;j+s) j=0i=m+1 m-sXt-jX s i+j+s = i2j2t-i-j (hi;j+s) j=0i=m+1 m-sXt-jX s i+j+s m-j-sXi-`X j+sk+`+j+s = i2j2t-i-j i2` 2i-k-` hk;`+j+s j=0i=m+1 `=0 k=m+1 m-sXt-jXm-j-sXi-`X s i+j+s j+sk+`+j+s = i2j2t-i-j i2` 2i-k-` hk;`+j+s: j=0i=m+1 `=0 k=m+1 So here the coefficient of ha;b+sis (upon setting k = a and ` + j = b, so that j ranges from 0 to b, and i ranges from a + b - j to t - j): Xb Xt-j s i+j+s j+s a+b+s i2j2t-i-j i2b-j2i-a-b+j: j=0i=a+b-j Hence this formula is indeed coassociative. |___| Since the E(m)'s are maximal quasi-elementary quotients, the following is an immediate corollary. Proposition 4.4.4.Under the coaction map lim-QHE**-! A lim-QHE**; we have bs+t-1_2cXt-jX s2i+j+s hts7-! i2jt-i-j hi;j+s: j=0 i=j+s+1 Dualizing, we have this formula for the action of A* on lim-HE**: 8 >ht-1;s+1if s + 1 < t - 1 and k = s, :0 otherwise: We give a graphical depiction of the (co)action in Figure 4.4.A, in which we in* *dicate the coaction by the primitives (i.e., the 2nth powers of 1). We end the section with a few examples. Example 4.4.5.Let R denote the ring lim-HE**. (a)The element ht;t-12 R in bidegree (1; |2t-1t|) = (1; 2t-1(2t- 1)) is an invariant, for t 1. Indeed, we know that h10 lifts to an element of the same name in ss1;1S0; also h421lifts to an element in ss4;24S0 (the element known as g or ___see [Zac67]). We do not know which power of ht;t-1 survives for t 3. 72 | 4. APPLICATIONS OF THE NILPOTENCE THEOREMS | | | | | _____.o.e.7 | h43 | @I | 2 | @ | @ | _____oe5 _____.o.e.6 | h32 h42 | @I @I | 1 1 | @ @ | @ @ | _____oe3_____oe4 _____.o.e.5 | h21 h31 h41 | @I @I @I | 0 0 0 | @ @ @ | @ @ @ | _____oe1_____oe2_____oe3 _____.o.e.4 | h10 h20 h30 h40 | _________________________________________________________| Figure 4.4.A. Graphical depiction of coaction of A on lim-HE**. k An arrow labeled by k representskan action by Sq2 2 A*, or equivalently a "coaction" by 21 2 A_in other words, a term of the form 2k1 (target) in the coaction on the source. (b)We have some families of invariants. h202 R is not invariant: we have h207-! 1 h20+ 21 h10: But since h10h21= 0 in R, then hi20hj21is invariant for all i 0 and j 1. It turns out that more of these elements lift to ss**S0 than one might expect from Theorem 3.1.2: the elements in the "Mahowald-Tangora wedge" [MT68 ] are lifts of the elements hi20hj21for all i 0 and j 8. (See [MPT71 ]; Zachariou [Zac67] first verified this for elements of the form h2i20h2(i+j)21for i; j 0.) These elements are distributed over a wedge be* *tween lines of slope 1_2and 1_5(in the Adams spectral sequence (t - s; s) gradin* *g). (c)Similarly, while h30and h31are not invariant, the monomials {hi30hj31hk32| i 0; j 0; k 1} are invariant elements. Hence some powers of them lift to ss**S0. We do not know what powers of them lift, but they will be distributed over a wedge between lines of slope 1_6and 1_27. Continuing in this pattern, we find th* *at for n 1, we have sets of invariant elements {hi0n0hi1n1:h:i:n-1n;n-1| i0; : :;:in-2 0; in-1 1}: The lifts of these elements lie in a wedge between lines of slope __1_2n-2* *and ____1_____ 2n-1(2n-1)-1. Hence the family of elements in the Mahowald-Tangora wedge is not a unique phenomenon_we have infinitely many such families, and when n 3 they give more than a lattice of points in ss**S0. (d)Margolis, Priddy, and Tangora [MPT71 , p. 46] have found some other non- nilpotent elements, such as x 2 ss10;63S0 and B212 ss10;69S0. These both 4.5. COMPUTATION OF A FEW BOUSFIELD CLASSES 73 come from the invariant z = h240h321+ h220h221h41+ h230h21h231+ h220h331 in R. While we do not know which power of z lifts to ss**S0, we do know that B21maps to the product h320h221z, and x maps to h520z. In other words, we have at least found some elements in the ideal (z) (R)A which lift to ss**S0. (e)Computer calculations have lead us to a few other such "sporadic" invariant elements (i.e., invariant elements that do not belong to any family_any family that we know of, anyway): h820h431+ h830h421+ h1121h31 in bidegree (12; 80), an element in bidegree (9; 104) (a sum of 8 monomial* *s in the variables hi;0and hi;1, 2 i 5), and an element in bidegree (13; 104)* * (a sum of 12 monomials in the same variables). We do not know what powers of these elements lift to ss**S0, nor are we aware of any elements in the * *ideal that they generate which are in the image of the restriction map from ss*** *S0. 4.5. Computation of a few Bousfield classes In [HPS97 , Section 5.1] we show that if a spectrum E is Bousfield equivalent to the sphere, then E detects nilpotence. This is true in any stable homotopy category, and it is not very deep. While we do not vouch for the depth of the nilpotence theorems of Chapter 3, we at least point out that they are not examp* *les of this generic nilpotence theorem. Many of the results in this section hold at all primes; some only hold at the prime 2. Unless otherwise indicated, fix an arbitrary prime p. W Theorem 4.5.1.We have > , and when p = 2, > E, where the wedge is taken over all quasi-elementary quotients E of A. The proof is quite similar to that of analogous results in [Rav84 ]. We need* * a few lemmas, first. Lemma 4.5.2.Suppose that B and C are quotient Hopf algebras of A that fit into a Hopf algebra extension B 2CFp,! B i C: (a)Suppose that dimFpB 2CFp= 1. If B and C are conormal quotient Hopf algebras of A, then [HC; HB]**= 0. Hence HC ^ IHB = 0; hence > . (b)Suppose that dimFpB 2CFp< 1. s (i)If B 2CFp contains some on or some pt with s < t, then > . (ii)Otherwise, if p = 2, then = . Proof. Part (a): The statement that [HC; HB]**= 0 implies the rest of the lemma, by Proposition 1.6.1, so we only have to verify that. For that verificat* *ion, we prove the corresponding statement about A*-modules, and let the reader translate back to A-comodules and to Stable(A). 74 4. APPLICATIONS OF THE NILPOTENCE THEOREMS Let B* be the dual of B, C* the dual of C, and B*==C* the dual of B 2CFp. We show that Ext**A*(A*==B*; A*==C*) ~=Ext**B*(Fp; A*==C*) = 0: By Lemma 4.5.3 below, as a B*-module, A*==C* is a direct sum of copies of B*==C* **, so it suffices to show that Ext**B*(Fp; B*==C*) = 0. From the Hopf algebra exte* *nsion C* -!B* -!B*==C*; we get a spectral sequence (as in Section 1.5) Ep;q2= ExtpB*==C*(Fp; ExtqC*(Fp; B*==C*)) ) Ext**B*(Fp; B*==C*): We claim that E**2= 0. Since B*==C* is a trivial C*-module, then Ext**C*(Fp; B*==C*) ~=B*==C* Ext**C*(Fp; Fp) as B*==C*-modules, and hence is free, and hence injective, over B*==C*. So Ep;** *2= 0 if p > 0, and M E0;*2= HomB*==C*(Fp; nffB*==C*): ff But Hom B*==C*(Fp; B*==C*) = 0 (see Lemma 4.5.4); this finishes the calculation. Part (b)(i) follows from Corollary 1.6.2 and induction, using the non-nilpot* *ence of the classes hts(when s < t), bts(when s < t), and vn. Part (b)(ii) is simila* *r,_but uses the nilpotence of htswhen s t: see Theorem B.2.1. |__| We have used the following two lemmas in the proof of Lemma 4.5.2. Lemma 4.5.3.Suppose B* and C* are normal sub-Hopf algebras of A*, with C* B*. Then as a B*-module, A*==C* is a direct sum of suspensions of B*==C*. Proof. We have isomorphisms of B*-modules (with B* acting on the left): A*C* Fp ~= A*B* B* C* Fp ~= A*==B* B*==C*: Now by normality, A*==B* is a trivial B*-module, so this tensor product is a di* *rect_ sum of copies of B*==C*, indexed by a vector space basis of A*==B*. |__| Lemma 4.5.4.Suppose B* and C* are normal sub-Hopf algebras of A*, with C* B*. Then ExtsB*==C*(Fp; B*==C*) = 0 for all s > 0. If dimFpB*==C* = 1, then HomB*==C*(Fp; B*==C*) = 0: Proof. B*==C* is self-injective, so the Ext group in question is zero if s > 0. So we need to show that Hom B*==C*(Fp; B*==C*) = 0, if B*==C* is infinite- dimensional. An element of this Hom group corresponds to an element x 2 B*==C* which supports no operations by elements of B*==C*; we want to show that any such x must be zero. Fix such an x, and assume that x 6= 0. Using the classification of (normal) sub-Hopf algebras of A* in Theorem 2.1.* *1, we see that there are two possibilities: either (1) B*==C* contains Pst's for a* *rbi- trarily large values of t, or (2) for some fixed t, B*==C* contains Pstfor all * *s 0. (By "B*==C* contains Pst," we mean that Pstis in B* and not in C*, so that it represents a nonzero class in the quotient.) In case (1), we choose Pstin B*==C* 4.5. COMPUTATION OF A FEW BOUSFIELD CLASSES 75 so that 2t> |x|. Then it is an easy exercise in multiplication with the Milnor * *basis to show that xPst6= 0. In case (2), we argue similarly to show that Pstx_6=_0 f* *or s 0. We leave the details to the reader. |__| (Perhaps Ext**B(k; B) should be zero for any graded connected Hopf algebra B over a field k, if B is nonzero in infinitely many degrees. We have not been ab* *le to show this, though.) Here are some consequences of Lemma 4.5.2(a). Corollary 4.5.5.[HD; S0]**= 0. Proof. Apply part (a) of the lemma with B = A and C = D. |___| Corollary 4.5.6.Let p = 2. Then [HE; HD]**= 0 for any quasi-elementary quotient E of A. Proof. This follows immediately from Lemma 4.5.2(a), since the maximal quasi-elementary quotient Hopf algebras of A are conormal quotients_of both A and D. |__| Corollary 4.5.7.[A; HD]**= 0. Hence if X is a finite object in loc(A), then X = 0. Proof. For the first statement, we apply the lemma with B = D and C = Fp. This then implies that [Y; HD]** = 0 for all Y in loc(A). If X is finite with Spanier-Whitehead dual DX, then [X; HD]**= HD**(DX). So it suffices to show that if X is finite and nontrivial, then HD**X 6= 0. Well, HD**X = 0 if and only if X is contractible when viewed as a cochain complex of comodules over D, in which case the homology of X must be zero. On the other hand, by the Hurewicz * * __ Theorem 1.4.4, if X is finite and nontrivial, then it has nonzero homology. * * |__| Proof of Theorem 4.5.1.It is clear (by Proposition 1.6.1(c), for instance) that we have _ E quasi- elem. (where the second inequality is only known to be valid when p = 2). By Corol- lary 4.5.5 and Proposition 1.6.1(f), we see that > ; in particular, HD* * ^ IS0 = 0. Hence the first inequality is strict. When p = 2, by Corollary 4.5.6 and Proposition 1.6.1(c)-(e), we see that HD ^ IHD 6= 0, while HE ^ IHD = 0 for all quasi-elementary E. Hence the_second inequality is strict. |__| W Remark 4.5.8.Let p = 2 and let X = HE. One can in fact show a stronger result_that X has no complement in HD. Suppose otherwise: suppose that there were an object G so that = _ and 0 = X ^ G. Smashing the former equality with IHD gives = _ ; But HE ^ IHD = 0 for all quasi-elementary E by Corollary 4.5.6 and Proposi- tion 1.6.1(b), so we have = : 76 4. APPLICATIONS OF THE NILPOTENCE THEOREMS Also, by Proposition 1.6.1(a), so we have = = <0>: But 6= 0. 4.6. Ideals and thick subcategories Let X be a finite spectrum; as in Section 4.1, we define the ideal I(X) HD** to be the radical of the kernel of HD**--^X--![X; HD ^ X]**: The periodicity theorem 4.1.3 tells us that for any element y in (HD**=I(X))A, * *we can lift some power of y to a central element in [X; X]**; therefore, the ideal* * I(X) is worth studying. In this section, we establish some properties of I(X)_finite generation and invariance_and we examine a possible relation with a classificat* *ion of thick subcategories of finite objects in Stable(A). 4.6.1. Ideals. Let p = 2. We can define the ideal I(X) for any object X in Stable(D); as we impose conditions on X_first finiteness, then X being defined over A rather than D_we find more properties of I(X). One of those properties is invariance: Definition 4.6.1.Suppose that R is a commutative comodule algebra over A (i.e., a commutative algebra and a left A-comodule, compatibly); we write for the comodule structure map. We say that an ideal IPof R is invariant under the A-coaction if for all x 2 I, (x) is of the form aj xj 2 A R, where each xj lies in I. Proposition 4.6.2. (a)Let X be a finite object of Stable(D). Then p_______________________ p ______________ ker(HD**-! [X; HD ^ X]**)= annHD**(HD**X): (b)Let X be a finite object of Stable(D). Then I(X) is a finitely generated i* *deal. (c)Let X be a finite object of Stable(A). Then I(X) is invariant under the A-coaction. The proof of part (a) is based on similar work in [Ben91b , Section 5.7]. Proof. Part (a): Let p _______________________ I = ker(HD**-! [X; HD ^ X]**); p ______________ J = annHD**(HD**X): The Yoneda action of HD** on HD**X factors through the composition action (since HD**is commutative). Hence I J. On the other hand, since X is finite, then it is in thick(S0). So any element of HD** that annihilates [S0; HD ^ X]** will also annihilate [X; HD ^ X]**. Hence J I. Part (b) [Sketch of proof]: To see that I(X) is finitely generated, note that since X is finite, then "most" of D acts trivially on X: we let B(n) denote the following quotient Hopf algebra of D: n+1 2n+2 B(n) = D=(1; 2; : :;:n) = F2[n+1; n+2; : :]:=(2n+1; n+2 ; : :):: 4.6. IDEALS AND THICK SUBCATEGORIES 77 Then B(n) should "act trivially on X" for n large enough. More precisely, there* * is an Atiyah-Hirzebruch spectral sequence with E2 = HFp**X HB(n)**) HB(n)**X: Since X is finite, then its homology HFp**X is bounded. So if n is large enough, then for degree reasons, the elements in the image of the edge homomorphism HFp**X -!E2 = HFp**X HB(n)00 are all permanent cycles. This is a spectral sequence of modules over HB(n)**, * *so everything must be a permanent cycle, and we find that HB(n)**X ~=HFp**X HB(n)**: Consider the ring map HD**-! HB(n)**. The annihilator in HD**of HD**X is contained in the annihilator in HD**of HB(n)**X. Since annB(n)**(HB(n)**X) = (0); then I(X) is contained in the radical of the kernel of HD**-! HB(n)**. We can calculate this kernel, up to radical, by imitating the arguments in Section 4.4* *; we find that we have an F-isomorphism HB(n)**-!F2[hts| s < t; n < t] = (htshvu | u t); and hence the radical of the kernel of HD**-! HB(n)**is equal to K = (hts| s < t n): If we view K as an ideal of the Noetherian ring F2[hts| s < t n]=(htshvu | u t) HD**; then we see that not only is K finitely generated, but so is any subideal of it. Part (c): Now we show that I(X) is invariant under the A-coaction. By Re- mark 1.3.8, both HD**and HD**X are A 2D Fp-comodules, and one can check that the action map HD** HD**X -!HD**X is a map of A 2D Fp-comodules.P Suppose that y 2 I(X), and that under the A 2D Fp-coaction, y maps to iai yi. (We may assume that the ai's are lin- early independent elements of A.) We want to show that each yi is in I(X); i.e., that some power of yiannihilates HD**X.k k Fix x 2 HD**X. If we assume that y2 x = 0, then we claim that y2ix = 0 for each i. Suppose that under the A 2D Fp-coaction, we have nX (4.6.3) x 7-! 1 x + bj xj: j=1 We prove, by induction on n, that y2kix = 0 for all i. When n = 0 (i.e., when x* * is primitive in HD**X), then we have HD**X -!(A 2D Fp) HD**X; k X 2k X 2k 2k 0 = y2 x7-! ( ai yi) (1 x) = ai yi x: i i Since the ai's are linearly independent, we conclude that y2kix = 0 for all i. 78 4. APPLICATIONS OF THE NILPOTENCE THEOREMS Suppose that the 2kth power of each yiannihilates every element of HD**which has at most n terms in its diagonal, and fix x with diagonal as in (4.6.3). The* *n by coassociativity, each xj has n terms or fewer in its diagonal, so we have k X 2k X 0 = y2 x7-! ( ai yi) (1 x + bj xj) i j X k k X k k = (a2i y2ix + a2ibj y2ixj) i j X k k = a2i y2ix: i Hence y2kix = 0 for each i. |___| 4.6.2. A thick subcategory conjecture. Since this subsection consists pri- marily of conjectures, we may as well let p be an arbitrary prime (although we * *have more evidence for the conjectures when p = 2). Theorems 3.1.2 and 4.1.3 provide support for several conjectures about the "global structure" of the category Stable(A). For instance, we have the followi* *ng suggested analogue of the result of Hopkins and Smith [HSb , Theorem 11], in which they identify the center of [X; X]* up to F-isomorphism, for any finite p- local spectrum X. Given a ring R, we let Z(R) denote the center of R. Conjecture 4.6.4.For any finite spectrum X, there is an F-isomorphism Z[X; X]**-! (HD**=I(X))A: We should point out there is not even an obvious map between these two rings. Theorem 4.1.3 also suggests a conjectured classification of thick subcategor* *ies of finite spectra in Stable(A); we spend most of this section discussing this conj* *ecture and related ideas. Conjecture 4.6.5.The thick subcategories of finite spectra in Stable(A) are * *in one-to-one correspondence with the finitely generated radical ideals of HD**whi* *ch are invariant under the coaction of A 2D Fp. As above, given a finite spectrum X, we let I(X) denote the radical of the k* *ernel of HD**-! [X; HD^X]**. The conjectured bijection should send an invariant ideal I to the full subcategory D(I) with objects {X finite| I(X) I}: This is clearly a thick subcategory. The other arrow in the bijection should se* *nd a thick subcategory D of the finite objects in Stable(A) to the ideal I(D), defin* *ed by " I(D) = I(X): X2obD This is a finitely generated radical invariant ideal by Proposition 4.6.2. Example 4.6.6.Here is a bit more evidence for Conjecture 4.6.5. (a)By Example 4.4.5(a), we have maps h10:S1;1-!S0; h421:S4;24-!S0: 4.6. IDEALS AND THICK SUBCATEGORIES 79 We let S0=h10 and S0=h421denote the cofibers of these. One can easily compute the ideals of these cofibers: p ____ I(S0=h10) = (h10); p ____ I(S0=h421) = (h21): Since h10h21= 0 in HD**, then Conjecture 4.6.5 would tell us that thick(S0=h10; S0=h421) = thick(S0): Let C = thick(S0=h10; S0=h421). One can show directly that S0 is contained in C, using the octahedral axiom: the cofiber of the map h10h421:S5;25-!S0 fits into a cofibration with S0=h10and S0=h421, and hence is in C. But sin* *ce h10h421is zero, then this cofiber is just S0 _ S4;25; hence S0 is in C. (b)By Example 4.4.5(b), we have non-nilpotent self-maps of the sphere spec- trum called hi20hj21, for certain exponentspi_and j. If i and j are both positive, then I(S0=(hi20hj21)) = (h20h21), so Conjecture 4.6.5 would im- ply that thick(S0=(hi20hj21)) is independent of i and j. Arguing as in part (a), one can see that this is true. (c)Lastly, we point out that thick(S0) 6= thick(S0=(hi20hj21)), even though the two spectra S0 and S0=(hi20hj21) have the same slope supports (Defi- nition 4.1.4). For instance, if we let d0 = h220h221, then S0=d0^ d-10S0 =* * 0, and hence X ^ d-10S0 = 0 for every X in thick(S0=d0). Here is a sketch of part of the proof of Conjecture 4.6.5. Conjecture 4.6.7.Given any invariant finitely generated radical ideal I of HD**, there is a finite spectrum X so that I(X) = I. Idea of proof.Suppose that I is generated by classes y1, y2, : :,:yn, with yi 2 HDsi;ti. We order these so that s1 s2 . . .sn; then (y1; : :;:yi) is invariant for each i n. For each i 0, we define spectra Xiinductivelypso_that I(Xi) = (y1; : :;:yi). We start by letting X0 = S0; then I(X0 ) = (0). Given Xi-1with I(Xi-1) = (y1; : :;:yi-1), Theorem 4.1.3 tells us that then Xi-1has a yi-map; we let Xi be the cofiber of this map. Clearly I(Xi) (y1; : :;:yi);_if * *we could prove equality, we would be done. |__| This would show that the composite I 7! D(I) 7! I(D(I)) is the identity. For the composite D 7! I(D) 7! D(I(D)), one needs to show that every thick subcategory D is of the form D = D(I) for some invariant ideal I. One might also conjecture that there is a bijection between the set of local* *izing subcategories of Stable(A) and the set of all radical ideals of HD**, but that * *seems a bit much to expect without any evidence. We end this section with one other body of ideas, based on work of Nakano and the author [NP ] (and this was based, in turn, on work of Friedlander and Parsh* *all [FP86 , FP87 ], among others). Let p = 2. We let W be the vector space W = SpanF2(Pst| s < t): We view W as being an inhomogeneous sub-vector space of the Steenrod algebra A*, and we let VD (F2) be the following subset of W: VD (F2) = {y 2 W | y2 = 0}: 80 4. APPLICATIONS OF THE NILPOTENCE THEOREMS We do not require that the elements y be homogeneous. One can show (as in the proof of [NP , 1.7]) that VD (F2) consists precisely of linear combinations* * of commuting Pst's_i.e., it is a union of affine spaces, one such space for each m* *aximal elementary quotient of A. Therefore it is equal to the prime ideal spectrum of lim-QHE**, and hence is homeomorphic to the prime ideal spectrum of HD**. At odd primes, we define Wev= SpanFp(Pst| s < t); Wodd= Span(Qn | n 0); and then let VD (Fp) be the set of all elements y = (y1; y2) in Wev Woddwith yp1= 0 and y22= 0. We do not have as nice a description of VD (Fp) at odd prime* *s; because of this, and for other technical reasons, it might be best to restrict * *the following discussion to the case p = 2. Given an element y 2 VD (Fp), we can construct the y-homology spectrum H(y), just as we did the Pst-homology spectrum in Definition 2.2.1 . Then given a spe* *c- trum X, we define its rank variety, VD (X), to be the following subset of VD (F* *p): VD (X) = {y 2 VD (Fp) | H(y)**X 6= 0}: We say that a subset V of VD (Fp) is realizable if V = VD (X) for some finite spectrum X. Conjecture 4.6.8.Let X be a finite spectrum. The rank variety VD (X) deter- mines the thick subcategory generated by X, and hence the ring of central self-* *maps of X, up to F-isomorphism. In other words, there is a bijection between the inv* *ari- ant finitely generated radical ideals of HD**and the realizable subsets of VD (* *Fp). This bijection should come about by the following: if X is a finite spec- trum, then there should be a homeomorphism (actually, an "inseparable isogeny" [Ben91b , p. 172]) between the prime ideal spectrum of HD**=I(X) and VD (X). 4.7.Construction of spectra of specified type Let p be a prime. In this section we construct certain objects for later use; these are analog* *ues of the "generalized Toda V (n) spectra," as used by Mahowald-Sadofsky [MS95 ], among others. Some of the basic ideas are standard; most of the rest are due to them. A few of the details are different in our setting. We make heavy use of Notations 2.2.4 and 2.4.1 in this section. The results of this section follow from Theorem 2.4.3 and Lemma 2.4.7, but they have the flavor of other results in this chapter; hence we include them he* *re. (In particular, the results holds at all primes, not just when p = 2.) We start by constructing the relevant spectra. See Notation 2.2.4 and 2.4.1 * *for the terminology used here. Proposition 4.7.1.Let p be a prime, and fix a slope n. Let 1 = d1 < d2 < . .<.dm be the slopes less than n. For any integers k1, : :,:km , there are int* *egers j1, : :,:jm with ki jifor each i, so that there is a spectrum F = F(uj1d1; : :;* *:ujmdm) satisfying the following. (a)When n = 1, F = S0. (b)F is Z(d)**-acyclic for d < n, and Z(n)**F 6= 0. 4.7. CONSTRUCTION OF SPECTRA OF SPECIFIED TYPE 81 (c)Hence F has a un-map; call it u. If Z(n)**(u) is multiplication by ujm+1n, then F(uj1d1; : :;:ujm+1n) is the fiber of u. More precisely, there is a * *fiber sequence jm+1 F(uj1d1; : :;:ujm+1n) -!F(uj1d1; : :;:ujmdm) un----!-jn;-njnF(uj1d1; : :;* *:ujmdm): (d)F is self-dual, as is its un-map u. That is, for some number q, we have DF ~=qF, and u maps to itself under the chain of isomorphisms [F; F] ~=[DF; DF] ~=[qF; qF] ~=[F; F]: Proof. The statements of (a)-(c) indicate how the spectra are constructed. Starting with S0, one applies Theorem 2.4.3 to find a u1-map uj11of it, and one lets F(uj11) be the fiber. By definition, essentially, uj11induces an isomorphi* *sm on Z(1)**, so F(uj11) is Z(1)**-acyclic; uj11induces zero on Z(d)**for d > 1, so F* *(uj11) is not Z(d)**-acyclic for any larger value of d. One proceeds inductively. Th* *is proves (a)-(c). Part (d) is also straightforward; it is proved by induction on m. We_leave t* *he details to the reader. |__| Up to suspension, the object F(uj1d1; : :;:ujmdm) is the analogue in Stable(* *A) of Mahowald and Sadofsky's spectrum (in the usual stable homotopy category) M(pj0; vj11; : :;:vjn-1n-1). We have used the letter F rather than M since our * *spectra are iterated fibers rather than cofibers, as in [MS95 ]. Also, the letter M can* * get somewhat overworked; in particular, we want to avoid confusion with the functor Mnfof Section 5.3. Now we establish the main properties of the spectra F(uj1d1; : :;:ujmdm). Theorem 4.7.2.Fix a prime p, a slope n, and other notation as in Proposi- tion 4.7.1, and let F = F(uj1d1; : :;:ujmdm) be as in that result. Then F satis* *fies the following properties. (a)For any finite spectrum W which is Z(d)**-acyclic for all d < n, W is in thick(F). (b)Hence the Bousfield class of F is independent of the choice of exponents j* *i. (c)Let u denote the un-map of F. Suppose that `1, : :,:`m are integers so that W = F(u`1d1; : :;:u`mdm) exists, and let v denote its un-map as given* * by Proposition 4.7.1(b). Then there are integers i and j so that ui^1W = 1F^vj as self-maps of F ^ W. (d)Suppose that `1, : :,:`m are integers so that F(u`1d1; : :;:u`mdm) exists,* * and so that ji `ifor each i. Then there is a map F(uj1d1; : :;:ujmdm) -!F(u`1d1; : :;:u`mdm) commuting with "projection to the top cell"_i.e., the map F(: :):-!S0. Proof. (a): For each i m, let F(Ui) = F(uj1d1; : :;:ujidi). We have cofiber sequences ujidi F(Ui) -!F(Ui-1) --! F(Ui-1): We claim that 1W ^ujidiis a nilpotent self map of W ^F(Ui-1). Once we know this, then we see that W ^ F(Ui-1) is in the thick subcategory generated by W ^ F(Ui) for each i. By induction, W ^ F(U0) = W is in the thick subcategory generated 82 4. APPLICATIONS OF THE NILPOTENCE THEOREMS by W ^ F(Um ), which is a subcategory of the thick subcategory generated by F(Um ) = F. The claim that 1W ^ujidiis nilpotent follows by application of the vanishing* * line theorem 2.3.1: the group [W ^ F(Ui-1); W ^ F(Ui-1)]**= ss**(W ^ DW ^ F(Ui-1) ^ DF(Ui-1)) has a vanishing line of slope n, and 1W ^ ujidiacts along a line of slope di< n. (b): This follows immediately from (a). (c): We would like to use the properties of y-maps discussed in Section 4.2,* * but these depend on the nilpotence theorem 3.1.5, and hence would force us to work * *at the prime 2. With a bit of care, we can avoid this dependence. By Proposition 4.7.1(b), F ^ W is Z(d)**-acyclic for d < n; hence [F ^ W; F ^ W]** has a vanishing line of slope at least n. If the slope is larger than n, t* *hen the powers of both u ^ 1W and 1F ^ v would eventually lie above the vanishing line, and hence would both be zero. So we may assume that the vanishing line has slope equal to n. By our choice of un-map u, Lemma 2.4.7 tells us that u ^ 1W is central in a band parallel to the vanishing line; 1F ^ v lies in that band, * *on the line of slope n through the origin. Therefore u ^ 1W and 1F ^ v commute. By our choice of maps u and v, after raising them to powers, we may assume that both of our un-maps agree when restricted to [F ^ W; HA(m) ^ F ^ W]**; for large enough m. But this ring is isomorphic to [F ^ W; F ^ W]**in the bideg* *ree of interest; hence our self-maps agree in [F ^ W; F ^ W]**. (d): This follows from (c). We prove it by induction on m. When m = 1, then the following diagram commutes: uj11 0 F(uj11)----!S0 ----! S : flfl ? fl ?yu`1-j11 u`11 0 F(u`11)----!S0 ----! S Hence there is an induced map F(uj11) -!F(u`11). Assume that we have a map F(uj1d1; : :;:ujm-1dm-1)f----!F(u`1d1; : :;:u`m-1dm-1): flfl fl fl flfl F W 4.7. CONSTRUCTION OF SPECTRA OF SPECIFIED TYPE 83 We abbreviate these spectra as F and W, as indicated. Consider the following diagram, in which the rows are fiber sequences: ujmdm F(ujmdm)----!F ----! F: flfl ? fl ?yu`m-jmdm u`mdm F(u`mdm)----!F ----! F ?? ? yf ?yf u`mdm W(u`mdm)----!W ----! W Clearly the top right square commutes. We claim that part (c) implies that the lower right square commutes. Given this, we get maps F(ujmdm) -!F(u`mdm) -!W(u`mdm): The composite is the desired map. To see that the lower right square commutes, we imitate the proof of Corol- lary 4.2.11. The key is to use Spanier-Whitehead duality so that the question is whether 1 ^ u`mdmand Du`mdm^ 1 agree as self-maps of the spectrum DF ^ W. By part (c), it suffices to show that F is self-dual (up to suspension), as is_its* * un-map u. But this is Proposition 4.7.1(d). |__| We point out that since F = F(uj1d1; : :;:ujmdm) is well-defined up to Bousf* *ield class, and since its un-map is essentially unique, then the telescope u-1nF is * *well- defined up to Bousfield class. But the telescope of a un-map of an arbitrary fi* *nite spectrum of type n could have a different Bousfield class. For example, at the * *prime 2, there is a non-nilpotent self-map of the sphere called d0: d0: S4;18-!S0: (Under the map ss**S0 -! HD**, the element d0 maps to h220h221_see Exam- ple 4.4.5(b).) Although S0 and S0=d0 are both type 0, they are not Bousfield- equivalent_S0 sees d-10S0, while S0=d0 does not. Hence the telescopes h-10S0 and h-10(S0=d0) probably have distinct Bousfield classes. We also point out that this theorem has the following obvious generalization, at least at the prime 2. See Conjecture 4.6.7 for related work. Theorem 4.7.3.Let p = 2, and fix a finitely generated invariant radical ideal I HD**. Write I = (y1; : :;:ym ), and order the generators so that (y1; : :;:y* *i) is invariant for each i m. For any integers k1, : :,:km , there are integers j* *1, : :,:jm with ki ji for each i, so that there is a spectrum F = F(yj11; : :;:yj* *mm) satisfying thepfollowing._ (a)When I = (0), then F = S0. (b)I is contained in the ideal of F. (c)Hence if y 2 HD**=I is invariant, then F has a y-map, u. (d)F is self-dual, as are its y-maps. (e)For any finite spectrum W with I I(W), then W is in thick(F). (f)Hence the Bousfield class of F is independent of the choice of exponents * *ji. (g)Fix y and u as in (c). For any finite spectrum W and any y-map v on W, there are integers i and j so that ui^ 1W = 1F ^ vj as self-maps of F ^ W. 84 4. APPLICATIONS OF THE NILPOTENCE THEOREMS (h)If `1, : :`:mare integers with `i ji and such that F(y`11; : :;:y`mm) exis* *ts, then there is a map F(y`11; : :;:y`mm) -!F(yj11; : :;:yjmm). 4.8. Further discussion: slope supports Let p be any prime. In this section, we discuss slope supports (Definition 4* *.1.4). This material was originally introduced in [Pal96b] as an approach to the perio* *d- icity theorem and thick subcategory conjecture. Since those topics now seem more closely related to I(X), the ideal of X, slope supports appear to be more perip* *h- eral. On the other hand, since Pst-homology groups do determine vanishing lines and other useful information, studying slope supports may be worthwhile. Our main result in this section is Proposition 4.8.1, which gives a classifi* *cation, when p = 2, of the possible supports of finite spectra. Recall from Definition 4.1.4 that Slopesis the set of slopes of A, whereas S* *lopes0 is the set ( Slopes0= {(t; s) | t > s 0}; p = 2; {(t; s) | t > s 0} [ {n | np 0};odd. There is, of course, a (meaningful) bijection between Slopesand Slopes0, with t* *he ps| 0 slope p|t_22 Slopescorresponding to (t; s) 2 Slopes, and when p is odd, |on| 2 Slopescorresponding to n 2 Slopes0. As in Definition 4.1.4, we say that the slope support of a spectrum X is the* * set supp(X) = {n | Z(n)**X 6= 0} Slopes: (See Notation 2.2.4 for the definition of Z(n).) We may also view supp(X) as be* *ing a subset of Slopes0, using the bijection above. We say that a subset T of Slope* *s0is admissible if T satisfies the following conditions: (i)When p = 2: (t; s) 2 T ) (t + 1; s) 2 T; (t; s) 2 T ) (t + 1; s - 1) 2 T; card(Slopes0\ T) < 1: (ii)When p is odd: the above conditions, as well as: n 2 T ) n + 1 2 T: We provide a bit of justification for the term "admissible" in the following. Proposition 4.8.1.[Pal96b, Prop. 3.10 and Thm. A.1] If T Slopes0is admissible, then T is the slope support of some finite spectrum. If p = 2, then* * the converse holds: if X is any finite spectrum, then suppX is admissible. We conjecture that the converse is also true when p is odd. We can at least * *prove the following: for X finite, if (Qn)**X = 0, then (Qn-1)**X = 0 (see [Pal96b, A.8]). We also note that there are no restrictions on the slope support of an arbitrary spectrum: the connective Pst-homology spectrum psthas slope support equal to {(s; t)}, so any subset of Slopes0may be realized as the support of a * *wedge of pst's (and similarly at odd primes, using the pst's and the qn's). 4.8. FURTHER DISCUSSION: SLOPE SUPPORTS 85 | | | | | | | s| @ | ________@ | T(t; s) | _________________________ | T(m) _________________________| _________________________|| t m Figure 4.8.A. T(t; s) and T(m) as subsets of Slopes0. Proof. We include a sketch of the proof that every admissible T is the slope support of some finite spectrum, since the corresponding result in [Pal96b] ass* *umed that p = 2. We focus on the case when p is odd; the reader can imitate this pro* *of or refer to [Pal96b] for the p = 2 case. Every admissible T Slopes0can be written as T = T(t1; s1) \ . .\.T(tn; sn) \ T(m) for some numbers ti; si; m, where T(t; s) is the largest admissible set not con* *taining (t; s)_i.e., it is the complement of {(t; s); (t - 1; s); (t - 1; s + 1); (t - 2; s); (t - 2; s + 1); (t - 2; s +* * 2); : :}:; and T(m) is the complement of {0; : :;:m}. See Figure 4.8.A. If we can find sufficiently nice finite spectra X(t; s) and X(m) so that suppX(t; s) = T(t; s)* * and suppX(m) = T(m), then we can realize any slope support T by the spectrum X(t1; s1)^: :^:X(tn; sn)^X(m). (One might worry that although Z(d)**X(ti; si) 6= 0 and Z(d)**X(tj; sj) 6= 0 for some d, one might have Z(d)**X(ti; si) ^ X(tj; s* *j) = 0. This does not happen for injective resolutions of finite comodules_see [NP , 1.5]_and it certainly does not happen for the examples we construct in the next paragraph. This is what we mean by "sufficiently nice.") We recall from [Mit85 ] that the quotient Hopf algebra n+1 pn p P(n) = A=(p1 ; 2 ; : :;:n+1; n+2; n+3; : :;:o0; o1; o2; : :): has the structure of an A-comodule, extending the P(n)-comodule structure. The same clearly holds for V (m) = E[o0; : :;:om ]: We will use P(n) and V (m) to refer to the Hopf algebras, the comodules, or the* *ir injective resolutions,sdepending on the context. It is well-known that H(P(n); * *Pst) is zero ifspt 6= 0 in P(n)_see Proposition 2.2.2 or [MP72 ]. On the other hand, if pt= 0 in P(n), then one can see by examining the Poincare series that H(P(n); Pst) 6= 0. So Z(d)**P(n) = 0 if and only if yd 6= 0 in P(n), and simila* *rly for V (m). Hence as subsets of Slopes0, we have suppP(n) = T(n + 1; 0); suppV (m) = T(m): 86 4. APPLICATIONS OF THE NILPOTENCE THEOREMS Also, there is a "pth power" functor on the category of A-comodules; if M is evenly graded, then M is defined to be ( (M)n = Mr n = pr; 0 n not divisiblepby; with comodule structure M :M -!A M given by M (x) = (F i) O M (x): Here, F is the Frobenius map on A defined by F(a) = ap, M is the comodule structure map on M, and i: M -!M is the map (which multiplies degrees by p) that sends y to y. One can readily see that if s < t, then suppsP(t - 1) = T(t; s): Hence an injective resolution of the comodule s1P(t1- 1) . . .snP(tn - 1) V (m) has slope support T = T(t1; s1) \ . .\.T(tn; sn) \ T(m): See [Pal96b, A.1] for the proof of the converse when p = 2. |___| 4.9. Further discussion: miscellany We have already mentioned several conjectures related to global structure of* * the category Stable(A) when p = 2_see Section 4.6. As far as working at odd primes, one has to prove analogues of Theorems 3.1.2 and 3.1.5 first; see Section 3.4 f* *or a discussion of those issues. We presented some preliminary computations of A-invariants in HD**in Sec- tion 4.4; it would be nice to have further results. Extensive computer calculat* *ions could be useful, and obviously it would be nice to find new families of invaria* *nts. Along similar lines, we could use more information about invariant ideals of HD* ***_ basic properties, examples, and of course a classification would be helpful. It is also natural to wonder if one can prove a version of the periodicity t* *heo- rem 4.1.3 which uses quasi-elementary quotients of A instead of D. CHAPTER 5 Chromatic structure In this chapter we discuss "chromatic" results in Stable(A). We start in Sec- tion 5.1 by discussing Margolis' killing construction [Mar83 , Chapter 21]. Th* *is is the analogue, in our setting, of the functor Lfnin the ordinary stable homot* *opy category. We give several different constructions of the functor, and we prove * *var- ious properties (e.g., for X with nice connectivity properties, if X has finite* * type homotopy, then so does LfnX). We also define an analogue of the functor Ln, and we show that Ln 6= Lfnif n > 1, at least at the prime 2. We have been using the functor H heavily throughout this book; the homotopy groups of HB, for B a quotient of A, are the cohomology groups of the Hopf alge* *bra B. In Section 5.2 we construct bH(-), a version of this functor whose homotopy groups are the Tate cohomology groups of B. The spectra bHA(m)turn out to be equal to LfnHA(m), for n sufficiently large compared to m; we use this result in Section 5.3 to prove that the "chromatic tower," the tower Lf0X - Lf1X - Lf2X - Lf3X - : :;: converges to X, if X is finite. (This is an extension of a theorem of Margolis [Mar83 , Theorem 22.1].) In Section 5.4 we discuss some other questions related to chromatic issues, * *such as constructing chromatic towers in different orders, and relating the chromatic tower construction to the multiple complex construction of Benson and Carlson. 5.1.Margolis' killing construction In this section we present Margolis' killing construction. Thissis a (smashi* *ng) localization functor that kills off Pst- and Qn-homology for ptand on of large * *slope. We make use of Notation 2.2.4 and 2.4.1 in this section. Warning. Note that un is an element of either Z(n)1;nor Z(n)2;2n, depending on the prime and the form of yn_see Proposition 2.2.2. So as to avoid dividing all of our arguments into cases, we will abuse notation and write uinfor the po* *wer of un in Z(n)i;*(and similarly for un-maps: a self-map uin:X -!X has bidegree (i; in)). Hence when yn = Pstand p is odd, only even powers of un make sense. We defined the notions of "thick" and "localizing" subcategories in Defini- tion 1.4.7. Definition 5.1.1.Recall from [Mil92] and [HPS97 , 3.3.3] that given any thick subcategory C of finite spectra, there is a functor LfC:Stable(A) -!Stabl* *e(A), called finite localization away from C, with the properties: (i)LfCis exact; when viewed as a functor LfC:Stable(A) -!LfCStable(A) to the category of LfC-local spectra, it has a right adjoint. 87 88 5. CHROMATIC STRUCTURE (ii)There is a natural transformation 1 -!LfC. (iii)LfCis idempotent_for any X, the map LfCX -! LfCLfCX induced by the natural transformation in (ii) is an equivalence. (iv)LfCis Bousfield localization with respect to the spectrum LfCS0. (v)For any X, LfCX = X ^ LfCS0. (vi)For any finite X, LfCX = 0 if and only if X 2 C; for any X, LfCX = 0 if a* *nd only if X is in the localizing subcategory generated by C. Properties (i)-(iii) say that LfCis a localization functor [HPS97 , 3.1.1]; * *prop- erty (v) says that LfCis smashing [HPS97 , 3.3.2]. We write CfCfor the corre- sponding acyclization functor; that is, CfCX is the fiber of X -!LfCX. Since Lf* *Cis smashing, then CfCX = X ^ CfCS0. Definition 5.1.2.For any slope n, let LfndenoteWfinite localization away from the thick subcategory of finite spectra which are dn Z(d)-acyclic, and write C* *fn for the corresponding acyclization. In Section 4.7, we constructed a finite spectrum F(uj11; : :;:ujnn) for an a* *ppro- priately chosen set of exponents j1, : :,:jn, indexed by slopes. We will also w* *rite F(Un) for this spectrum. If d is the next largest slope after n, then by constr* *uction, F(Un) has a ud-map. Let Tel(d) denote its telescope. Theorem 4.7.2 tells us that the Bousfield class is independent of the exponents j1, : :,:jn; as not* *ed after that theorem, the same is true of . The following is based on Mahowald and Sadofsky's analysis in [MS95 ] of the functor Lfnin the ordinary stable homotopy category. Proposition 5.1.3.The functor Lfncan be described in any of the following ways: (a)as finite localization away from the thickWsubcategory generated by F(Un); (b)as Bousfield localization with respect to dn Tel(d); (c)and as a colimit:_if we let F(ui11; : :;:uinn) -!S0 be projection to the t* *op cell with cofiber F(ui11; : :;:uinn), then the map X -!LfnX is given by __ i i X -!lim-!i1;:::;inX ^ F(u11; : :;:unn): (The maps in the direct system will be defined in the proof.) W On the other hand, for n large, Lfnis not Bousfield localization with respec* *t to dn Z(d). See Proposition 5.1.10. W Proof. By Theorem 4.7.2(a), the thick subcategory of finite dn Z(d)-acyclic spectra is equal to the thick subcategory generated by F(ui11; : :;:uinn) for a* *ny choice of exponents i1; : :;:in. This proves part (a). Part (b) is essentially a Bousfield class computation. Repeated application * *of Proposition 1.6.1(a) gives us (5.1.4) = _ _ . ._.; as well as pairwise orthogonalityWof the Bousfield classes on the right. We nee* *d to show thatWa spectrum X is dn Tel(d)-acyclic if and only if it is Lfn-acyclic. * *Assume that dn Tel(d)**X = 0. By the decomposition of Bousfield classes (5.1.4), we see that = , so LfnX = 0 if and only if LfnX ^ F(Un) = 0. But 5.1. MARGOLIS' KILLING CONSTRUCTION 89 certainly LfnF(Un) = 0, so since Lfnis smashing,Wthen LfnF(Un) ^ X = 0. To show the converse, it suffices to show that F(Un) is dn Tel(d)-acyclic; this follow* *s by the orthogonality of the above Bousfield classes. To prove (c), we first need to construct the direct system. By "projection to the top cell," we mean the composite F(Un) -! : :-:!F(U1) -! F(U0) = S0. Using properties of the spectra F(Uk) and their uj-maps (Theorem 4.7.2), we can choose the uj-maps compatibly; i.e., if we have exponents ij and `j so that ij * *`j and F(ui11; : :;:uinn) and F(u`11; : :;:u`nn) are defined, then there is a map F(ui11; : :;:uinn) -!F(u`11; : :;:u`nn) __ i which commutes with projection to the top cell. We let F(u11; : :;:uinn) denote the cofiber of F(ui11; : :;:uinn) -! S0, and we define the spectrum L0nX to be * *the following colimit: __ i X ----! lim-!F(u11; : :;:uinn) ^ X: fl i1;:::;in fl flfl flfl 0 X --g--! L0nX 0 For example, if n = 1, then we have S0 g-!u-11S0, where u-11S0 is the mapping telescope of u1: S0 -!S0, and fiber(g0) = F(u11) is the analogue of the mod p1 Moore spectrum. We need to verify that L0nagrees with Lfn. To do this, we note that L0nis smashing, and we show that g0:S0 -! L0nS0 is an LfnS0-equivalence and that L0nS0 is LfnS0-local. By construction, the fiber of g0is in the localizing subc* *ategory generated by the spectra F(ui11; : :;:uinn), and hence g0is an LfnS0-equivalenc* *e. To show that L0nS0 is local, we have to show that [W; L0nS0]**= 0 for any Lf* *n- acyclic W. For any finite localization, the acyclics are the localizing subcate* *gory generated by the finite acyclics, so it suffices to show this when W is a finit* *e acyclic. Note that if W is a finite acyclic, then so is DW, the Spanier-Whitehead dual of W. Since [W; L0nS0]**= [S0; DW ^ L0nS0]**= ss**L0nDW; it suffices to show that L0nW = 0 for any finite acyclic W. This is easy: if W * *is an acyclic, then for each slope j n, if k is the next largest slope after j, t* *hen by vanishing lines, every uk-map of F(ui11; : :;:uijj) ^ W is nilpotent. Theref* *ore,_a cofinal set of maps in the direct system defining L0nW is zero. |* *__| W Recall from Definition 2.4.2 that a spectrum is of type n if it is d n. In other words, (i)0Z(d)**CfnX = 0 if d n, and (ii)0Z(d)**LfnX = 0 if d > n. (b)Suppose that X is a CL-spectrum (Definition 1.4.6). Then LfnX is "bounded to the left": for each i, ssijLfn= 0 for j 0. If, in addition, X has fini* *te type homotopy, then so does LfnX. Proof. Part (a): By the theory of finite localization in [HPS97W], the spect* *rum CfnX is in theWlocalizing subcategory generated by the finite dn Z(d)-acyclic* *s, and hence is dn Z(d)-acyclic itself. This proves (i). Since homology commutesWwith direct limits,Wwe see that Z(d)**Tel(k) is zero if d 6= k; in particular, kn Tel(k) is d>n Z(d)-acyclic. Since Lfnis Bousfi* *eld 5.1. MARGOLIS' KILLING CONSTRUCTION 91 W W localization with respect to kn Tel(k), then we conclude that LfnS0 is d>nZ(d* *)- acyclic. Since Lfnis smashing, this proves (ii). Part (b): Given X as in the statement, we show that the maps in the direct system defining LfnX (Proposition 5.1.3(c)) are isomorphisms on ss** in a range increasing with the exponents. As above, we let F(Uk) = F(ui11; : :;:uikk) and we assume that d is the next largest slope after k. We write F(Uk)(uidd) = F(ui11; : :;:uidd) for the fiber * *of the ud-map on F(Uk). Consider the following diagram: uidd : :-:---! -id+1;-iddF(Uk)----! F(Uk)(uidd)----!F(Uk)----! : : : ?? ? fl yuidd ?yf1 flfl u2idd : :-:---!-2id+1;-2iddF(Uk)----!F(Uk)(u2idd)----!F(Uk)----! : : : ?? ? fl yuidd ?yf2 flfl u3idd : :-:---!-3id+1;-3iddF(Uk)----!F(Uk)(u3idd)----!F(Uk)----! : : : ?? ? fl yuidd ?yf3 flfl .. . . . .. .. It suffices to show that the maps fr^ 1X are isomorphisms in a range increasing with r. Well, fr^ 1X is an isomorphism whenever the map uidd^1X-(r+1)i +1;-(r+1)i d (5.1.7) -rid+1;-riddF(Uk) ^ X -----! d F(Udk) ^ X from the left column is. The fiber is -rid+1;-riddF(Uk)(uidd) ^ X; this spectrum has a vanishing line with some slope ` > d, and we can use Remark 2.3.4 to comp* *ute its intercept. To do this, we compute (HFp)**F(Uk)(uidd)_each uj-map induces the zero map on (HFp)**, so this computation is easy. We find that (i)(HFp)i*= 0 for i 0, (ii)(HFp)i*= 0 for i > 0, and (iii)(HFp)*j= 0 for j < N for some fixed number N. By the Hurewicz Theorem 1.4.4, (i) and (iii) also hold for ss**; hence we can c* *ompute the equation of the vanishing line for F(Uk)(uidd). Smashing with X moves the intercept a bit, so we find that ssijF(Uk)(uidd) ^ X = 0 when j < `i + N0 for s* *ome N0 which is independent of r. Therefore the map (5.1.7)is an isomorphism on ssij when j + ridd < `(i + rid- 1) + N0, i.e., when j < `i + rid(` - d) + (N0- `). S* *ince ` > d, then as r increases, this range increases. So we see that for each s, then the graded groups sss*of the direct system h* *ave a uniform bound to the left; hence the same is true of the direct limit. If X a* *lso has finite type homotopy, then since the homotopy of each F(Uk) is of finite ty* *pe, the same goes for F(Uk) ^ X; since the homotopy of the direct system stabilizes* *_at each bidegree, we are done. |__| Remark 5.1.8. (a)Much of this was already known in the module setting. For p = 2, this is due to Margolis [Mar83 , Theorem 21.1]; part (a) (and t* *he connectivity in part (b)) for arbitrary primes can be found in [Pal92, The- orem 3.1]. Given an A-comodule M and a slope n, then the module-version 92 5. CHROMATIC STRUCTURE of the killing construction (dualized to the category of comodules) gives * *co- modules M and M<1; n>, well-defined up to injective summands, and an injective comodule J so that the following is short exact: 0 -!M -!M J -!M<1; n> -!0: Being well-defined up to injective summands translates in our setting to being well-defined up to an object of loc(A) (cf. Lemma 5.1.9). So if X is an injective resolution of M, then an injective resolution for M<1; n> is a connective cover for LfnX. (b)By studying the proof of Theorem 5.1.6(b), one can show that CfnS0 has a nice vanishing curve: for i 0, ssijCfnS0 has a vanishing line of slope n,* * and if d is the slope preceding n, then for i < 0, ssijCfnS0 has a vanishing l* *ine of slope d. For example, when n = 1, we have Cf1S0 ----! S0 ----! Lf1S0: flfl fl fl fl flfl flfl F(u11)----! S0 ----! u-11S0 So ss**Lf1S0 = u-11ss**S0 = h-110ss**S0 = F2[h110]: Hence ssijCf1S0 has a vanishing line of slope 2 for positive i, and a vani* *shing line of slope 1 (a vertical line, in (i; j - i)-coordinates) for i negativ* *e. See also Figure 5.1.A. (c)One can generalize this construction. We set p = 2 and use the periodicity theorem 4.1.3 and its corollary Theorem 4.7.3. Let I be a finitely generat* *ed invariant ideal of HD**, and let C be the thick subcategory of finite spec* *tra X with I(X) I. Then we have a finite localization functor LfIand a cofibration CfIX f-!X g-!LfIX: If the ideal I is generated by classes udi, then Z(d)**f is an isomorphism for d 62 {di}, and Z(d)**g is an isomorphism if d 2 {di}. The analogue of Proposition 5.1.3 holds. Our proof of Theorem 5.1.6(b), on the other hand, does not work in this situation; it may be that these finite type and connectivity results only hold when I is as in the theorem. We need the following property of Lfnin Section 5.2. Lemma 5.1.9.For any slope n, LfnA = 0. Hence any spectrum X in the local- izing subcategory generated by A is Lfn-acyclic. Proof. Since the category of Lfn-acyclics contains a finite spectrum_F(Un), then it contains A by Lemma 1.4.9. |__| We close this section with a few remarks about the telescope conjecture (see [Rav84W] and [HPS97 , 3.3.8]). Let Ln denote Bousfield localization with respect to dn Z(d). Proposition 5.1.10.Let p = 2. We have L1 = Lf1. If n > 1, then Ln 6= Lfn. 5.1. MARGOLIS' KILLING CONSTRUCTION 93 | i| slope _1_ | n-1 | | ii | | i i i | i|i? | i i ___________________________________________||ii | j - i ___|__slopeo_1_e | d-1 | | | | Figure 5.1.A. "Vanishing curve" for ssij(CfnS0): ssij(CfnS0) is zero above the two indicated lines. Here, d is the slope preceding n, and slopes are labeled in (i; j - i)-coordinates. These lines are drawn through the origin for convenience, but their intercepts may actually be nonzero. One can interpret the statement "Ln = Lfn" as an analogue of the telescope conjecture, so this result would say that the telescope conjecture fails except* * when n = 1. On the other hand, it seems more proper to refer to the statement "every smashing localization functor is a finite localization" as the telescope conjec* *ture [HPS97 , 3.3.8]. Since we do not know whether Ln is smashing, this result does not necessarily present a counterexample to this version of the telescope conje* *cture. Proof. We know that the sphere has a u1-map; a simple computation shows that u-11S0 = Z(1), so that Bousfield localization with respect to Lf1S0 = u-11* *S0 is the same as that with respect to Z(1). There is a non-nilpotent element d0 2 ss4;18S0 = Ext4;18A(F2; F2). We claim * *that d-10S0 is Ln-acyclic for all n, and Lfn-local for n 3. Since 3 is the next slo* *pe after 1, this covers all of the bases. For degree reasons, d0 induces the zero map on Z(d)**for all d. (If an eleme* *nt ff 2 ssijS0 is nonzero on Z(d)**S0 = Z(d)**, then we must have d = j=i.) Hence Z(d)**(d-10S0) = 0 for all d, and d-10S0 is Ln-acyclic for allWn. To show that * *d-10S0 is Lfn-local, we show that [F; d-10S0]**= 0 for any finite dn Z(d)-acyclic F. * *We compute: [F; d-10S0]**= [S0; DF ^ d-10S0]** = [S0; d-10DF]** = d-10ss**DF: W Since F is a finite dn Z(d)**-acyclic, so is DF (by Proposition 2.2.6). By The- orem 2.3.1, ss**DF has a vanishing line of some slope m > n. Since n 3, we know that m 6, so the slope m is larger than 9_2, the slope of d0. Hence d0 ac* *ts nilpotently on ss**DF; i.e., d-10ss**DF = 0. |___| When p is odd, one can again show that L1 = Lf1. There is every reason to expect that there are non-nilpotent elements in Ext**A(Fp; Fp) which are not de* *tected by any single Z(d); once one knew this, one could conclude that Ln 6= Lfnfor n large. 94 5. CHROMATIC STRUCTURE 5.2.A Tate version of the functor H In this section we introduce a "Tate" version of the functor H, and we relate it to the functor Lfn, at least when applied to the quotient Hopf algebra A(m).* * We will use our computations here to prove chromatic convergence in Section 5.3. Let B be a quotient Hopf algebra of A. Note that practically all of our resu* *lts hold in the category Stable(B); the main exceptions are the strict inequalities* * in Theorem 4.5.1. Let resA;B:Stable(A) -! Stable(B) denote the forgetful functor (also known as restriction); resA;Bhas a right adjoint, induction, written indB* *;A, and defined by X 7! A 2BX (cf. Lemma 1.3.4(a)). These functors are exact (i.e., they take cofibrations to cofibrations), and restriction preserves the smash pr* *oduct and the sphere object. (In the language of [HPS97 , Section 3.4], the restricti* *on functor is a stable morphism.) When B is a finite-dimensional Hopf algebra, we may consider another sta- ble homotopy category associated to it, called the stable category of B-comodul* *es, written StComod(B). We provide a brief review of the relevant results here; see [HPS97 , Section 9.6] for a few more details. The objects of StComod(B) are B- comodules, and the morphisms Hom_*Bare defined as follows: define the morphisms of degree zero to be Hom_B(X; Y ) = Hom B(X; Y )= ', where f ' g :X -! Y if f - g factors through an injective comodule. Hence two comodules are equiva- lent in StComod(B) if they differ by injective summands. Define the desuspension functor -1 by the short exact sequence 0 -!X -!B X -!-1X -!0; where B X is the cofree comodule on X (Definition 1.1.9). This functor is invertible: X is any comodule which fits into a short exact sequence 0 -!X -!I -!X -!0; where I is injective. (Since B is finite, then injectives and projectives are t* *he same, and there are enough projectives.) We let Hom_iB(X; Y ) = Hom_B(iX; Y ): StComod(B) is a stable homotopy category. Indeed, on Stable(B) one has fi- nite localization (Definition 5.1.1) away from the thick subcategory generated * *by the finite spectrum B = HFp; StComod(B) is equivalent to the full subcategory of Stable(B) of LfB-local objects (see [HPS97 , 9.6.3-4]). So we have a functor LfB:Stable(B) -!StComod(B) with a right adjoint J :StComod(B) -!Stable(B). (Every localization functor has a right adjoint, namely inclusion of the local * *objects into the category.) Definition 5.2.1.We note that the object HB is indB;A(S0), and we define the object bHB to be indB;A(J(S0)). Fix a slope n. On each of the categories Stable(A), Stable(B), and StComod(B* *), we can define the functor Lfnto be smashing with LfnS0, or with its image under 5.2. A TATE VERSION OF THE FUNCTOR H 95 resor LfBO res. This gives us the commuting diagram of functors LfB Stable(A) --res--!Stable(B)----! StComod(B); ?? ? ? y Lfn ?yLfn ?yLfn LfB f LfnStable(A)-res---!LfnStable(B)----!LnStComod(B) along with the commuting diagram of their right adjoints Stable(A) -ind---Stable(B)---J- StComod(B): x? x x ? RStable(A) ??RStable(B) ??RSt(B) LfnStable(A)ind----LfnStable(B)J----LfnStComod(B) (The first diagram commutes by the definition of the vertical functors, and the second diagram commutes as a result_given a commuting square of left adjoints, their right adjointssalso commute.) Recall from Notation 2.2.4 that ydis the el* *ement of A_either ptor on_"with slope d." Proposition 5.2.2.Fix a quotient Hopf algebra B of A, and let N = max{d | yd 6= 0 inB}: Then if n is a slope larger than N, we have LfnHB = bHB: Proof. We claim that the two maps ff: HB -!LfnHB; fi :HB -!HbB; are the same. To show this, we show that the spectrum bHB is Lfn-local, and that the map fi :HB -!HbB is an Lfn-equivalence. By Lemma 5.2.4 below, the fiber of fi has homotopy concentrated in the third quadrant, so by Lemma 1.4.8, it is in loc(A). In particular, it is Lfn-acyclic by Lemma 5.1.9; hence fi is an Lfn-equ* *ivalence. To show that bHB is local, we show that it is in the image of the right adjo* *int RStable(A)of Lfn:Stable(A) -! LfnStable(A). We claim, in fact, that RSt(B)is the identity functor. Given this claim, then we use the commutativity of the diagram of right adjoints: we have HbB = ind(JS0) = ind(J(RSt(B)S0)) = RStable(A)(ind(JS0)): Hence bHB is in the image of RStable(A), and so it is local. It remains to verify the claim that RSt(B)is the identity functor, or what i* *s the same, that Lfn:StComod(B) -!StComod(B) is the identity functor. So it suffices to show that the cofiber of S0 -!LfnS0 * *is zero in the category StComod(B). This cofibration is obtained by applying the functor 96 5. CHROMATIC STRUCTURE ||| ||| s|_|| | | s|_|| | | |_|| | | |_|| | | |_|| | | |_|| | | |_|| | | |_|| | | |_|| | |_|| | |__|| |__|| _____________________|||||||||||||||||_||____________||||||||||||||||||||||* *|_|| |__ t - s | |__ t - s |_ | |_ |_ | |_ |_ | | |_ |_ | | |_ | | | | HA(1)st bHA(1)st Figure 5.2.A. The coefficients of HA(1)and bHA(1)when p = 2. Since the top cell of A(1) is in degree 6, the third quadrant of HbA(1)**is the same as the first quadrant, reflected across the origin and then translated by (-1; -6). LfBO resto this cofibration in Stable(A): CfnS0 -!S0 -!LfnS0: So the cofiber under consideration is LfB(resCfnS0). By Proposition 5.1.3, LfnS* *0 is finite-localization away from the thick subcategory generated by F(Un); but F(U* *n) has no Z(d)-homology for any d with yd 2 B, so res(F(Un)) is in loc(B). Since res(F(Un)) is finite, it is in fact in thick(B). By fundamental properties of f* *inite localizations (Definition 5.1.1), res(CfnS0) is in loc(F(Un)), and hence in loc* *(B). Hence LfB(resCfnS0) = 0, as desired. |___| We are particularly interested in the case B = A(m) (this quotient of A is defined in Example 2.1.4). Corollary 5.2.3.Fix an integer m 0. For n sufficiently large, we have LfnHA(m) = bHA(m): In particular, n should be larger than ( m+1 max{d | yd 6= 0 inA(m)} = |m+1|p=_2m+1 - 1if p = 2, 2|m | = p - pif p is odd. Now we "compute" the homotopy groups of HbB. See Figure 5.2.A for an example. Lemma 5.2.4.Let B be a finite-dimensional quotient Hopf algebra of A, and consider ssijbHB. (a)When ij < 0, then ssijbHB = 0. (In other words, the homotopy is concen- trated in the first and third quadrants.) (b)When i and j are nonnegative, the map HB -!HbB induces an isomorphism ssijbHB ~=ssijHB. (c)Let d be the maximal degree in which B is nonzero. When i and j are negative, we have an isomorphism ssijbHB ~=ss-1-i;-d-jHB. 5.3. CHROMATIC CONVERGENCE 97 Proof. We work in the category Stable(B). We have adjoint functors L: Stable(B) -!StComod(B); J :StComod(B) -!Stable(B); and we want to compute the homotopy of JS0 by comparing it to that of S0 2 Stable(B). To do this, we need to recall the description of the functor J from [HPS97 , 9.6.7]. Let I denote an injective resolution of the B-comodule Fp(i.e., I ~=S0 in Stable(B)); then Hom Fp(I; Fp) is a projective resolution of Fp. We s* *plice these resolutions together to get the "Tate complex" tB(Fp): : :-:!Hom(I1; Fp) -!Hom (I0; Fp) -!I0 -!I1 -!: ::: Then the functor J is defined by J(M) = tB(Fp) M. It is clear that the map S0 -! J(S0) induces an isomorphism on ssi*for i 0. If we take I to be a minimal injective resolution, then Hom (I; Fp) is a minimal projective resoluti* *on; by minimality, [S0; JS0]i*= [Fp; JS0]i*gives the primitives in JS0 in degree i,* * and there is one primitive for each summand isomorphic to B. Since Hom (B; Fp) ~=__ -dB, we get the reflection and translation as described in (c). |_* *_| 5.3. Chromatic convergence It is easy to see that given a spectrum X and slopes n1 and n2 with n1 < n2, then the map X -! Lfn1X factors through X -! Lfn2X. Hence we get a tower of cofibrations: .. . . . .. ..: ?? fl ? y flfl ?y Cf3X ----! X ----! Lf3X ?? fl ? y flfl ?y Cf2X ----! X ----! Lf2X ?? fl ? y flfl ?y Cf1X ----! X ----! Lf1X (Strictly speaking, we have defined Lfnonly when n is a slope. For a general n, we define Lfn, as inWDefinition 5.1.2, to be finite localization away from the * *finite spectra which are dn Z(d)-acyclic. So if n is not a slope, then Lfn= Lfn-1.) We may as well just focus on the right hand column, giving the following diagram: 0---- Lf1X ---- Lf2X ---- Lf3X ---- : ::: x x x =?? ?? ?? M1fX M2fX M3fX Here, MnfX = X ^ MnfS0 is defined to be the fiber of LfnX -!Lfn-1X. We call this diagram the chromatic tower for X. (Clearly if n is not a slope, then MnfX = 0.) The following theorem, in the module setting for p = 2, is due to Margolis [Mar83 , Theorem 22.4] (see also [Pal94]). 98 5. CHROMATIC STRUCTURE Theorem 5.3.1 (Chromatic convergence).If X is finite, then X = lim-nLfnX. Indeed, the proof shows that the tower of groups ss**LfnX is pro-constant. In order to prove Theorem 5.3.1, we need the following proposition. This proposition may hold for general spectra, not just for finite spectra and rings* *, but we do not know how to prove that. Z(n) is defined in Notation 2.2.4. Proposition 5.3.2.Suppose that n is a slope. If R is either a finite spectrum or a ring spectrum, and if Z(n)**R = 0, then MnfR = 0. Proof. If X is a finite spectrum, then X ^ DX is a ring spectrum; arguing as in Proposition 2.2.6, we have Z(n)**X = 0 , Z(n)**(X ^ DX) = 0 and MnfX = 0 , Mnf(X ^ DX) = 0. So the finite case follows from the ring spectrum case. Suppose that m is the slope preceding n. Then we have a cofiber sequence __ MnfR -!LfnR H-!LfmR: By the octahedral axiom, we also have a cofiber sequence CfnR H-!CfmR -!MnfR: We start by looking at the map H, or rather, the maps related to it in the dire* *ct limit defining LfnR. Consider the following diagram. (As in Section 5.1, F(Um ) denot* *es F(ui11; : :;:uimm), and F(Um )(uinn) denotes the fiber of the uinn-map on F(Um * *).) in^1R F(Um ) ^ R----! F(Um )(uinn) ^-Rh---!F(Um ) ^ Run-----!F(Um ) ^ R: ?? ? fl ? y uinn^1R ?y flfl uinn^1R?y 2in^1R F(Um ) ^ R----!F(Um )(u2inn) ^-Rh---!F(Um ) ^-Run----!F(Um ) ^ R: The un-map uinnon F(Um ) is adjoint to a un-element ff 2 ss**F(Um ) ^ DF(Um ). So there are two un-elements in the ring spectrum (F(Um ) ^ DF(Um )) ^ R: ff ^ * *1R and 1F(Um)^DF(Um)^ 0 = 0. They obviously commute, so Lemma 4.2.7 tells us that some power of ff ^ 1R is null. If we write j :S0 -!R for the unit map of R, then ff ^ 1R is adjoint to uinn^ j :F(Um ) -!F(Um ) ^ R, and the self-map uinn^* * 1R factors through this map; hence some iterate of it is null as well. So for a large enough exponent in, the map h is a map onto a wedge summand. Furthermore, the vertical maps uinn^ 1R in the above diagram are null. To get information about H :LfnR -!LfmR, we map everything into R and take cofibers; we see that the cofiber of h is still a map onto a direct summand, an* *d the__ vertical maps are still null. Taking direct limits gives the desired isomorphis* *m. |__| Corollary 5.3.3.Fix m. For n sufficiently large, MnfHA(m) = 0; hence the chromatic tower for HA(m) stabilizes. Proof. This follows from the previous result and Proposition 2.2.5. |__* *_| In particular, at the prime 2, MnfHA(m) = 0 if n > |m+1|; at odd primes, MnfHA(m) = 0 if n > p|m|_2. Note that by Corollary 5.2.3, the chromatic tower f* *or HA(m) converges to bHA(m), not to HA(m). So while we can compute the limit of the chromatic tower here, we also see that we do not have chromatic convergence in this case. 5.4. FURTHER DISCUSSION 99 Proof of Theorem 5.3.1.If X is finite, then smashing with X commutes with inverse limits. Since LfnX = X ^ LfnS0, then chromatic convergence for S0 implies chromatic convergence for all finite spectra. By Theorem 5.1.6(b), LfnS0 has finite type homotopy bounded to the left, so by Proposition 2.1.5, we see that LfnS0 = lim-mHA(m) ^ LfnS0 = lim-mLfnHA(m): Hence we have lim-nLfnS0= lim-n(lim-mLfnHA(m)) = lim-m(lim-nLfnHA(m)) = lim-mbHA(m): Now we use Lemma 5.2.4 to compute the homotopy groups of lim-mbHA(m): in the first quadrant, the inverse limit stabilizes in each bidegree to the homoto* *py of lim-mHA(m) = S0, by Proposition 2.1.5. In the second and fourth quadrants, there is no homotopy at any stage in the inverse system. In the third quadrant, for each bidegree (i; j) there is an m0 so that ssijbHA(m)= 0 for all m m0; so the inverse limit has no homotopy in the third quadrant, either. Hence_the_map S0 -!lim-mbHA(m)is an isomorphism in homotopy. |__| One could probably get another proof of chromatic convergence by making precise, and then using, the result mentioned in Remark 5.1.8(b). By the way, Margolis gets chromatic convergence in his setting for all bound* *ed below modules; since we do not have convergence for HA(n)(which is an injective resolution of a bounded below comodule), we cannot expect our result to general- ize, precisely as stated, to other spectra. On the other hand, perhaps one could modify the proof of Theorem 5.3.1 to show that for any CL-spectrum X, then X -!lim-LfnX is a connective cover. For this, the alternate formulation of CL a* *fter Definition 1.4.6 might be useful_X is CL if and only if there are numbers i0, i* *1, and j0 so that X has a cellular tower built of spheres Si;jwith i0 i i1 and j - i j0. 5.4.Further discussion It seems most convenient to consider the chromatic tower as above, in which we kill off the Pst-homology groups of X in order of slope. This ordering is wh* *at allows us to prove chromatic convergence, via the "connectivity result" of Theo- rem 5.1.6(b). But at the prime 2, at least, we can kill off the generators of H* *D** in many different orders: we can start with any invariant element in HD**, take its cofiber, and continue (as in Theorem 4.7.3 and Remark 5.1.8). Can one still prove convergence? Does other structure reveal itself when one works with other generators of HD**or with other orderings of the same generators? As noted after the statement of Proposition 5.1.10, the validity of the stro* *ng form of the telescope conjecture is not knownWin Stable(A). If one could show that Bousfield localization with respect to dn Z(d) were smashing, that would disprove it; we would be interested in any progress along these lines. 100 5. CHROMATIC STRUCTURE It is conceivable that the finite-type result of Theorem 5.1.6(b) could have bearing on the convergence (in the ordinary stable homotopy category) of the to* *wer : :-:!Lf2X -!Lf1X -!Lf0X: After all, the vn-localized Adams spectral sequence in [MS95 ] has an E2-term w* *ith some relation to what we call ss**LfnS0, so whether one can prove convergence t* *his way, computations and other information about ss**LfnS0 could have applications to the finite localization functor Lfnin ordinary stable homotopy theory. Work of Mahowald and others has led to calculations similar to that of bHA(m) in Lemma 5.2.4. This may provide more connections between our functor Lfnand topology. Chromatic convergence for the sphere gives a convergent filtration of ss**S0* * = Ext**A(Fp; Fp). When p = 2, Mahowald and Shick [MS87 ] give another convergent filtration; are these filtrations the same? The slope of the element n+1 is 2n+* *1- 1; this element corresponds to the periodicity operator vn = hn+1;0, which has slo* *pe 2n+1-1. Mahowald and Shick also construct something they call v-1nExt**A(F2; F2* *). How does this compare to ss**(Lf2n+1-1S0)? (Shick [Shi88] has done similar work at odd primes, and one can naturally ask the same questions about that.) In parallel with the study of chromatic phenomena in the ordinary stable hom* *o- topy category, one should try to understand the filtration pieces in the chroma* *tic tower, and other "monochromatic" objects: objects X so that Mnf= X. Other than the trivial case of M1fX = h-110X, we have no information about these obje* *cts. Lastly, it seems possible that the chromatic tower constructed here (as well as those with other orderings) are the Steenrod algebra analogues of the multip* *le complexes of Benson and Carlson [BC87 ]. Is this a good analogy? If so, does it give any new insight into Stable(A), Stable(kG), or Stable() for an arbitrary commutative Hopf algebra ? (Evens and Siegel [ES96 ] have extended the multi- ple complex construction to modules over finite-dimensional cocommutative Hopf algebras, or equivalently to comodules over finite-dimensional commutative Hopf algebras, so their work is relevant here.) APPENDIX A Two technical results A.1. An underlying model category Let be a graded commutative Hopf algebra over a field k. In this section we (briefly) describe a model category whose associated homotopy category is equiv- alent to Stable(). The main results here (Theorems A.1.3 and A.1.4) are due to Hovey; see [Hov97 ] for the details. Recall that Quillen [Qui67 ] defined the notion of a closed model category; * *this is a category, like the category of topological spaces, in which one has a noti* *on of a well-behaved homotopy relation between maps. This allows one to define a new category, the associated homotopy category, which has the same objects as t* *he original one, but where the morphisms are the homotopy classes of morphisms in the original category. (Briefly, a closed model structure is determined by spec* *ifying three classes of maps_weak equivalences, fibrations, and cofibrations_satisfying certain properties. See [Qui67 ], as well as [DS95 ] and [Hov97 ], for details.) These days, by the way, one often says "model category" rather than "closed model category." Model categories are useful because, while one can do many constructions wor* *k- ing entirely in a homotopy category, for certain more delicate operations one n* *eeds to work at the "point-set" level_i.e., in the model category. For example, while Adams' definition in [Ada74 ] of the homotopy category of spectra is extremely useful, there are constructions one cannot do unless one has a model category u* *n- derlying it. Hence today one has various definitions of model categories of spe* *ctra, each with its own advantages and disadvantages. In our work with the Steenrod algebra, we have not needed a model category underlying Stable(). Nonetheless, it seems like a good idea, pedagogically and * *for future applications, to set up such a model category. We assume that is a graded commutative Hopf algebra over a field k. We let Ch() denote the category whose objects are cochain complexes of -comodules (not necessarily injective ones), and whose morphisms are cochain maps. We put a model category structure on Ch(); to do this, we need to specify the weak equivalences, the fibrations, and the cofibrations in Ch(). Notation A.1.1.Given cochain complexes X and Y , let [X; Y ] denote the set of cochain homotopy classes of maps from X to Y . Given a -comodule M, let SiM denote the cochain complex which is M in degree i, and zero elsewhere. Let L(k) denote an injective resolution of the trivial module k, and let S denote t* *he set of simple comodules of . For i an integer, M 2 S , and X any object of Ch(), we define ssi(X; M) (the "ith homotopy group of X with coefficients in M") by ssi(X; M) = [SiM; L(k) X]: 101 102 A. TWO TECHNICAL RESULTS We say that a map f :X -!Y in Ch() is a weak equivalence if and only ssi(f; M) is an isomorphism for all integers i and all simple comodules M. We say that a * *map f :X -! Y is a fibration if and only if each component fn: Xn -!Yn of f is an epimorphism with injective kernel. We say that a map f :X -!Y is a cofibration if and only if each component fn: Xn -!Yn is a monomorphism. Remark A.1.2.One can show (see [Hov97 ]) that a map f is a weak equiva- lence if and only if 1L(k) f is a cochain homotopy equivalence. Also, note that ssi(X; k) = [Sik; L(k) X] is the ith homology group of the cochain complex of primitives of L(k) X; this is a useful alternate description of ss*(-; k). Theorem A.1.3.[Hov97 ] With weak equivalences, fibrations, and cofibrations defined as above, Ch() is a model category. Its associated homotopy category is equivalent to Stable(). (Note that with this model structure, every object of Ch() is cofibrant.) One can in fact give Ch() the structure of a cofibrantly generated model cat- egory, as follows; we refer to [Hov97 ] for the proofs. Given a -comodule M, we let DnM denote the (contractible) cochain complex which is M in degrees n and n + 1, zero elsewhere, with differential given by the identity map: : :-:!0 -!0 -!M -1M-!M -!0 -!0 -!: ::: Both Sn (defined in Notation A.1.1) and Dn are functors from Ch() to itself. We let J denote the following set of maps in Ch(): J = {Dnf | n 2 Z; f :M ,! N an inclusion of finite-dimensional}comodules: We define I by I = J [ {Sn+1M ,! DnM | n 2 Z; M simple}: (Here the map Sn+1M ,! DnM is the obvious inclusion.) Theorem A.1.4.[Hov97 ] With I and J defined as above, Ch() has the struc- ture of a cofibrantly generated model category, in which I is the set of genera* *ting cofibrations and J is the set of generating trivial cofibrations. A.2. Vanishing planes in Adams spectral sequences In this appendix, we describe a result of Hopkins and Smith [HSa ]: for any nice ring spectrum E and any number m, the collection of spectra X so that the E-based Adams spectral sequence converging to ss*X has a vanishing line of slope m at some Er-term forms a thick subcategory. We also give a convergence result for the Adams spectral sequence in Stable(A). We work in a stable homotopy category like Stable(A)_one in which homotopy groups are bigraded, and cofibrations look like this: : :-:!1;0Z -!X -!Y -! Z -!-1;0X -!: ::: Hence the Adams spectral sequence is trigraded, so we discuss vanishing planes rather than lines. (In a short subsection below, we also give the original stat* *ement due to Hopkins and Smith_the statement of the corresponding theorem in the ordinary stable homotopy category.) A.2. VANISHING PLANES IN ADAMS SPECTRAL SEQUENCES 103 Recall that we presented the Adams spectral sequence in this setting in Theo- rem 1.5.1. We defined "generic" in Definition 1.4.7. If a spectrum W is (w1; w2* *)- connective (Definition 1.4.3) but neither (w1+ 1; w2)-connective nor (w1; w2+ 1* *)- connective, we write kWk = (w1; w2). Theorem A.2.1.Suppose that E is a spectrum satisfying the conditions given before Theorem 1.5.5, and consider the E-based Adams spectral sequence E****(X) ) ss***(X): Fix numbers m 0 and n. The following properties on a spectrum X are each generic. (i)There exist numbers r and b so that for all s, t, and u with s m(s + u) + n(t + u) + b; we have Es;t;ur(X) = 0. (ii)There exist numbers r and b so that for all finite spectra W with kWk = (w1; w2) and for all s, t, and u with s m(s + u - w1) + n(t + u + w2) + b; we have Es;t;ur(X ^ W) = 0. Note that the "slope" (m; n) of the vanishing plane is fixed, but the interc* *ept b and term r of the spectral sequence may vary in these generic conditions. Remark A.2.2. (a)We want E to be a nice ring spectrum so we can iden- tify the E2-term and so we have some convergence information. For the proof of the theorem, convergence is important, but the form of the E2-term is not. Hence, if we can guarantee convergence by some other means, then we can discard the assumptions on E. For example, in our application of this genericity result in Chapter 3, we know that the Adams spectral se- quence coincides with the spectral sequence associated to a Hopf algebra extension (Proposition 1.5.3), and hence has good convergence properties. (In this application, it is also easy to verify the conditions mentioned i* *n the theorem.) (b)We assume that m 0 for convenience in stating and proving Lemma A.2.4 below. It is certainly possible that the result holds regardless of the va* *lue of m. One proves this theorem by showing that the purported generic conditions are equivalent to conditions on composites of maps in the Adams tower; then one sho* *ws that those conditions are generic. We start by describing a_construction of the Adams spectral sequence. Given a ring spectrum E, we let E denote the fiber of the unit map S0 -!E. For any integer s 0, we let __^s FsX = E ^ X; __^s KsX = E ^ E ^ X: 104 A. TWO TECHNICAL RESULTS We use these to construct the following diagram of cofibrations, which we call * *the Adams tower for X: X _______F0X --g-- F1X --g-- F2X --g-- : ::: ?? ? ? y ?y ?y K0X K1X K2X This construction satisfies the definition of an "E*-Adams resolution" for X, as given in [Rav86 , 2.2.1]_see [Rav86 , 2.2.9]. Note also that FsX = X ^ FsS0, and the same holds for KsX_the Adams tower is functorial and exact. Given the Adams tower for X, if we apply ss**, we get an exact couple and hence a spectral sequence. This is called the E-based Adams spectral sequence. More precisely, we let Ds;t;u1= sss+u;t+uFsX; Es;t;u1= sss+u;t+uKsX: If we let g :Fs+1X -! FsX denote the natural map, then g* = sss+u;t+u(g) is the map Ds+1;t+1;u-11-!Ds;t;u1. Then we have the following exact couple (the triples of numbers indicate the tridegrees of the maps): (-1; -1; 1) Ds;t;u1_________________Ds+1;t+1;u-11oe @ @ (0; 0;@0)@ (1; 0; 0) @@R Es;t;u1 This leads to the following rth derived exact couple, where Ds;t;uris the image* * of gr-1*, and the map Ds+1;t+1;u-1r-!Ds;t;uris the restriction of g*: (-1; -1; 1) Ds;t;ur_________________Ds+1;t+1;u-1roe @ @ (r - 1; r - 1; -r@+@1) (1; 0; 0) @@R Es+r-1;t+r-1;u-r+1r Unfolding this exact couple leads to the following exact sequence: (A.2.3) : :-:!Es;t+1;u-1r-!Ds+1;t+1;u-1r-!Ds;t;ur-!Es+r-1;t+r-1;u-r+1r-!::: : Fix numbers m 0 and n. With respect to the E-based Adams spectral sequence E****(-), we have the following conditions on a spectrum X: (1)There exist numbers r and b so that for all s, t, and u with s m(s + u) + n(t + u) + b, the map gr-1*:sss+u;t+u(Fs+r-1X) -!sss+u;t+u(FsX) is zero. A.2. VANISHING PLANES IN ADAMS SPECTRAL SEQUENCES 105 (2)There exist numbers r and b so that for all s, t, and u with s m(s + u) + n(t + u) + b, we have Es;t;ur(X) = 0. (3)There exist numbers r and b so that for all finite spectra W with kDWk = (-w1; -w2) and for all s with s mw1 + nw2 + b, then the composite W -!Fs+r-1X -!FsX is null. (Here, DW denotes the Spanier-Whitehead dual of W.) (4)There exist numbers r and b so that for all finite spectra W with kWk = (w1; w2) and for all s, t, and u with s m(s + u - w1) + n(t + u - w2) + b, we have Es;t;ur(X ^ W) = 0. Each condition depends on a pair of numbers r and b, and we write (1)r;bto mean that condition (1) holds with the numbers specified, and so forth. Lemma A.2.4.Fix numbers m 0, n, r 1, and b. We have the following implications: (a)(1)r;b) (2)r;b+r-1. (b)(2)r;b) (1)r;b-m. (c)(3)r;b) (4)r;b+r-1. (d)(4)r;b) (3)r;b-m (Obviously, (3)r;b) (1)r;band (4)r;b) (2)r;b, but we do not need these facts* *.) We assume that r 1 so that we have the inequality r 1 + m. Since r is the term of an Adams spectral sequence, assuming that r 1 is not much of a restriction. Proof. As above, we write g for the map Fs+1X -!FsX and g* for the map Ds+1;t+1;u-11-!Ds;t;u1, so that Ds;t;uris the image of gr-1*:sss+u;t+uFs+r-1X -!sss+u;t+uFsX: (a): Assume that if s m(s + u) + n(t + u) + b, then gr-1*:sss+u;t+u(Fs+r-1X) -!sss+u;t+u(FsX) is zero; i.e., that Dstur= 0. If s m(s + u) + n(t + u) + b, then since r m, we have s + r m((s + r) + (u - r + 1)) + n((t + r - 1) + (u - r + 1)) + b. So we * *see that Ds+r;t+r-1;u-r+1r= 0. By the long exact sequence (A.2.3), we conclude that Es+r-1;t+r-1;u-r+1r= 0 when s m(s + u) + n(t + u) + b. Reindexing, we find that Ep;q;vr= 0 when p - r + 1 m(p + v) + n(q + v) + b; i.e., condition (2)r;b* *+r-1 holds. (b): If Es;t;ur(X) = 0 when s m(s + u) + n(t + u) + b, then (since r - 1 m) Es+r-1;t+r-2;u-r+2r(X) = 0 also. So by the exact sequence (A.2.3), we see that Ds+1;t;ur-!Ds;t-1;u+1ris an isomorphism under the same condition. This map is induced by g*: sss+1+u;t+uFs+1X -! sss+u+1;t+uFsX, so we conclude that when s m(s + u) + n(t + u) + b, we have lim-qsss+u+1;t+uFqX = Ds;t-1;u+1r; lim-1qsss+u+1;t+uFqX = 0: But by convergence of the spectral sequence, we know that lim-qsss+u+1;t+uFqX =* * 0, so Ds;t-1;u+1r= imgr-1*= 0. Reindexing gives Dp;q;vr= 0 when p m(p + v - 1) + n(q + 1 + v - 1) + b; i.e., (2)r;bimplies (1)r;b-m. Parts (c) and (d) are similar: 106 A. TWO TECHNICAL RESULTS (c): Fix a finite spectrum W with kWk = kD(DW) k = (w1; w2). By condition (3), we see that the composite s+u;t+uDW -!Fs+r-1X -!FsX is null when s m(s + u - w1) + n(t + u - w2) + b. This is adjoint to Ss+u;t+u-!Fs+r-1(X ^ W) -!Fs(X ^ W): Hence Dstur(X ^ W) = 0 when s m(s + u - w1) + n(t + u - w2) + b, in which case Ds+r;t+r-1;u-r+1r(X ^ W) is also zero. So by the long exact sequence (A.2.* *3), Es+r-1;t+r-1;u-r+1r(X ^ W) = 0 when s m(s + u - w1) + n(t + u - w2) + b. Reindexing as in part (a) gives us condition (4)r;b+r-1. (d): Fix a finite spectrum W with kDWk = (-w1; -w2). Condition (4) tells us that Es;t;ur(X ^ DW) = 0 when s m(s + u + w1) + n(t + u + w2) + b. By the exact sequence (A.2.3), we see that Ds;t;ur(X ^ DW) = 0 under the same condition (as in part (b)), so the composite Ss+u;t+u-!Fs+r-1(X ^ DW) -!Fs(X ^ DW) is null when s m(s + u + w1) + n(t + u + w2) + b - m. Hence the composite S0 -!Fs+r-1(X ^ DW) -!Fs(X ^ DW) is null when s mw1+ nw2+ b - m, as is W -!Fs+r-1X -!FsX; by Spanier-Whitehead duality. So (4)r;bimplies (3)r;b-m. |___| It is easy to prove Theorem A.2.1, once we have the lemma. Proof of Theorem A.2.1.The proofs of the genericity of the two statements are similar, so we only prove that condition (i) is generic. We know by Lemma A.2.4 that condition (i) is equivalent, up to a reindexing, to (*)There exist numbers r and b so that for all s, t, and u with s m(s + u) + n(t + u) + b, the map gr-1:Fs+r-1X -!FsX is zero on sss+u;t+u. This is generic, by the usual sort of argument: since the Adams tower is functo* *rial, if Y is a retract of X, then the Adams tower for Y is a retract of the Adams to* *wer for X. So if Fs+r-1X -! FsX is zero on sss+u;t+u, then so is Fs+r-1Y -! FsY . (Given Ss+u;t+u-!Fs+r-1Y , then consider Ss+t;t+u----! Fs+r-1Y ----! FsY ?? ? yi ?yi Fs+r-1X ----! FsX ?? ? yj ?yj Fs+r-1Y ----! FsY Since sss+u;t+uFs+r-1X -! sss+u;t+uFsX is 0, then the map Ss+u;t+u-!FsX is null. But Ss+u;t+u-!FsY factors through this map, and hence is also null.) A.2. VANISHING PLANES IN ADAMS SPECTRAL SEQUENCES 107 Given a cofibration sequence X -!Y -! Z in which X and Z satisfy conditions (*)r;band (*)r0;b0, respectively, we show that Y satisfies (*)r+r0-1;max(b;b0-* *r+1). Consider the following commutative diagram, in which the rows are cofibrations: Fs+r+r0-2X----! Fs+r+r0-2Y ----! Fs+r+r0-2Z ?? ? ? y ?yff ?yfi Fs+r-1X ----! Fs+r-1Y ----! Fs+r-1Z ?? ? ? yfl ?yffi ?y FsX ----! FsY ----! FsZ We assume that s m(s + u) + n(t + u) + max(b; b0- r + 1), so that we have s m(s + u) + n(t + u) + b; s + r - 1 m(s + u) + n(t + u) + b0: If we apply sss+u;t+uto this diagram, then sss+u;t+ufi = 0; hence sss+u;t+uff f* *actors through sss+u;t+uFs+r-1X. Since sss+u;t+ufl = 0, though, then sss+u;t+u(ffi O f* *f)_= 0. This shows that condition (*), and hence condition (i), is generic. * *|__| We end this section with a convergence result. We say that a spectrum X is E-complete if the inverse limit of the Adams tower for X is contractible. Proposition A.2.5.We work in Stable(A). Suppose that E is a ring spectrum as in Theorem A.2.1. Then every connective spectrum X is E-complete. Proof. This is an easy connectivity result: our conditions on E ensure that if X is connective, then FsX is (is; js)-connective, where is increases with s.* * In particular, for all i and j, ssijFsX = 0 for s large enough; hence the inverse_* *limit of the Adams tower will have no homotopy, and the lim1term will be zero. |__| Note that this applies when E = HB, for B any quotient Hopf algebra of A. A.2.1. Vanishing lines in ordinary stable homotopy. We give the state- ment of the original Hopkins-Smith result in ordinary stable homotopy theory. Given a connective spectrum W, we write |W| for its connectivity. Theorem A.2.6 ([HSa ]).Suppose that E is a ring spectrum as in [Rav86 , 2.2.5], and consider the E-based Adams spectral sequence E***(X) ) ss*(X). Fix a number m 0. The following properties on a spectrum X are each generic. (i)There exist numbers r and b so that for all s and t with s m(t - s) + b, we have Es;tr(X) = 0. (ii)There exist numbers r and b so that for all finite spectra W with |W| = w and for all s and t with s m(t - s - w) + b, we have Es;tr(X ^ W) = 0. The proof above can be easily modified to apply here. The same proof (in the case m = 0) also shows the following (using the langu* *age of [Chr97 ]). Corollary A.2.7.If I is an ideal of maps that is part of a projective class, then the following property is generic: 108 A. TWO TECHNICAL RESULTS oThere exist numbers r and b so that for all s b, the composite gr-1:Fs+r-1X -!FsX is in I. APPENDIX B Steenrod operations and nilpotence in Ext **(k; k) Let A be the dual of the mod p Steenrod algebra. In this appendix we recall a few results about Steenrod operations in the cohomology of any Hopf algebra , and we focus in particular on the quotient Hopf algebras B of A. Then we discuss the nilpotence of certain classes in Ext**B(Fp; Fp), which we need to p* *rove the nilpotence theorems of Chapter 3. B.1.Steenrod operations in Hopf algebra cohomology In this short section we recall a few facts about Steenrod operations in Hopf algebra cohomology. May's paper [May70 , Section 11] is our basic reference; see also [Sin73], [Wil81 ], [Saw82 ], [BMMS86 ], and [Rav86 ] for related results. Suppose that is a Hopf algebra over the field Fp. Then there are Steenrod operations acting on Ext**(Fp; Fp): (a)If p = 2, then for each n 0 there are the following operations: fSqn:Exts;t(F2; F2) -!Exts+n;2t(F2; F2): (b)If p is odd, then for each n 0 there are the following operations (where q = 2p - 2): fPn: Exts;2t(Fp; Fp) -!Exts+qn;2pt(Fp; Fp); fifPn: Exts;2t(Fp; Fp) -!Exts+qn+1;2pt(Fp; Fp): Note that fifPn must be treated as a single operation, not the composite of two operations. These satisfy the usual properties of Steenrod operations: the Cartan formula, * *the Adem relations, and an instability condition: if x 2 Exts;t(Fp; Fp) (with s and* * t even if p is odd), then fSqs(x) = x2;p = 2; fP s_2(x) =pxp;odd: Note that at odd primes, the operations are zero on classes in Exts;t(Fp; Fp) when t is odd. This is an artifact of the grading conventions on the operations. To remedy this, one can either use a different grading convention (see [May70 ] and [BMMS86 , IV.2] for two different conventions), or one can define operati* *ons indexed by half-integers, as in [Rav86 , 1.5.2]. Example B.1.1. (a)Fix a Hopf algebra . Recall from Lemma 1.1.15 that Ext1 is isomorphic to the vector space of primitives in . If x is primitiv* *e, then we write [x] for the corresponding element of Ext. We have the follow* *ing 0 Steenrod operation: fP0[x] = [xp]. (At the prime 2, fSq[x] = [x2].) One 109 110 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k) 0 can compute fP0and fSqon general Ext classes by a similar formula_these operations are induced by the following map on the cobar complex: [x1|x2| : :|:xn] 7! [xp1|xp2| : :|:xpn]: See [May70 ] for a proof. (b)If = Fp[x]=(x2) with x primitive, then Ext**(Fp; Fp) ~=Fp[h], where h = 1 n [x]. If p = 2, then fSq(h) = h2, and fSq(h) = 0 for n 6= 1. If p is odd (in which case |x| must be odd), then all operations vanish on h (because of o* *ur grading conventions). (c)If p is odd and = Fp[x]=(xp) with x primitive, then Ext**(Fp; Fp) ~=E[h] Fp[b], where h = [x] and b is the p-fold Massey product of h with itself. * *We have fifP0(h) = b, and all other operations on h are zero; also, fP1(b) = * *bp, and all other operations on b are zero. B.2. Nilpotence in HB**= Ext**B(F2; F2) In this section we recall a result of Lin which we use in the proofs of the * *main results of Chapter 3. Lin's result first appeared as [Lin, 3.1 and 3.2]. Wilker* *son [Wil81 , 6.4] has also proved a related result. Let A be the dual of thesmod 2 Steenrod algebra. Recall from Notation 1.3.9 and Remark 2.1.3sthat if 2tis primitive in a quotient Hopf algebra B of A, then we let hts= [2t] denote the corresponding element of HB1;*. Theorem B.2.1.Suppose that n1 2n2 B = A=(21 ; 2 ; : :): is a quotient Hopf algebra of A so that, for some integer m, we have ni< 1 for i = 1; 2; : :;:m - 1. Fix so that 2m is primitive in B. (a)If m, then hm; is nilpotent in HB**. (b)Fix an integer ` m, and suppose that for some , 2` is primitive in B. If ` , then h`;hm; is nilpotent in HB**. (Part (a) is a corollary of part (b)_just set ` = m and = .) For example, the class h1 = h112 Ext1;2A(F2; F2) is nilpotent. This is easy * *to show directly: the (reduced) diagonal 2 7! 211in A gives the relation h11h10= 0 1 in Ext**A(F2; F2). Applying the Steenrod operation fSqgives h12h210+ h311= 0; so multiplying through by h11 and using h11h10 = 0 yields h411= 0. Note that 0 applying fSqto this gives h41n= 0 for all n 1. For our purposes, part (a) is one of the key ingredients in the proof that r* *e- striction to the quotient Hopf algebra D detects nilpotence. Part (b) is used * *in the classification of quasi-elementary quotients of A, essentially, and is also* * used in showing that restricting to the quasi-elementary quotients detects nilpotence. B.3. NILPOTENCE IN HB**= Ext**B(Fp;Fp) WHEN p IS ODD 111 B.3. Nilpotence in HB**= Ext**B(Fp; Fp) when p is odd In this section we discuss the odd-primary analogue of Theorem B.2.1. Fix an odd prime p, and let A besthe dual of the mod p Steenrod algebra. Recall from Remarks2.1.3 that if ptis primitive in a quotient Hopf algebra B of A, then hts= [pt] is the corresponding element of HB1;*= Ext1;*B(Fp; Fp), and bts2 HB2;*is defined to be fifP0(hts). (Alternatively, btsis equal to the p-fold Massey product of htswith itself.) If on is primitive in B, we let vn = [on] be* * the corresponding element of HB1;*. For convenience, we restate Conjecture 3.4.1. This would be the odd-primary analogue of Theorem B.2.1(a). Conjecture B.3.1.Fix integers s and t. Suppose that n1 pn2 e e B = A=(p1 ; 2 ; : :;:o00; o11; : :): s is a quotient Hopf algebra of A in which pt is nonzero and primitive. If s t, then btsis nilpotent in HB**. By the way, one could consider a partial analogue of Theorem B.2.1(b): under conditions on s, t, and n, the product btsvn is nilpotent. Proposition B.3.2.Fix integersss, t, and n. Suppose that B is a quotients Hopf algebra of A in which pi= 0 for i < t, and oj = 0 for j < n; hence ptand on are primitive in B. If n s, then btsvn is nilpotent in Ext**B(Fp; Fp). Proof. The coproduct n+tXi on+t7! pn+t-i oi+ on+t 1; i=0 together with the conditions on B, gives the following relation in the cobar co* *mplex for B: n+t-1X hn+t-i;ivi= 0: i=n Pn+t-1 Hence htnvn = - i=n+1hn+t-i;ivi. Applying Steenrod operations gives the fol- lowing (here one needs to use half-integer indexed operations fP n_2, or one ne* *eds to use operations Pk which are indexed differently): n+t-1X btnvpn= bn+t-i;ivpi; i=n+1 2 n+t-1X p2 bt;n+1vpn= bn+t-i;i+1vi ; i=n+1 .. . s+n-1 n+t-1X ps+n-1 bt;svpn = bn+t-i;i+s-nvi : i=n+1 s s+n-1 __ Since pi= 0 for i < t, then we see that btsvpn = 0. |__| 112 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k) Another analogue of Theorem B.2.1(b) would be that under conditions on s, t, v, and u, the product btsbvu is nilpotent. Conjecture B.3.1 should be a spec* *ial case of such a result, and since we do not know how to prove this conjecture, we will not be able to prove such an analogue of Theorem B.2.1(b). For the remainder of this section, we provide more evidence and partial resu* *lts towards Conjecture B.3.1. s+1 We point out that if s < t, then Fp[t]=(pt ) is a quotient Hopf algebra of B; the cohomology of this quotient is E[ht0; : :;:hts] Fp[bt0; : :;:bts]: The element btsis non-nilpotent when restricted to this quotient, and hence non- nilpotent in ExtB. When s t, though, surprisingly little seems to be known abo* *ut the nilpotence (or lack thereof) of bts. For example, while it is easy to verif* *y that h411= 0 in Ext**A(F2; F2) at the prime 2 (see Section B.2), we have not been ab* *le to locate or prove a similar result for b11at an arbitrary odd prime. Working at the prime 3, Nakamura [Nak75 ] proved that b211= h11z for some z. Since h11is in odd total degree, then h211=s0; hence b411= 0. s Note that if an element pt0is primitive in a Hopf algebra B, then so is ptfor any s s0. s Lemma B.3.3.Fix a Hopf algebra B, and fix integers s0 t 1. Suppose that pt0is primitive in B. (a)If bt;s0is nilpotent, then so is bt;s0+1. s0-t ps0-t (b)Conversely, assume that p1 = . .=.t-1 = 0 in B, and that s0 t. If bt;s0+1is nilpotent, then so is bt;s0. As an application of (a), Nakamura's calculation implies that b41;n= 0 for a* *ll n 1, when p = 3. One might conjecture (based on very little evidence) that at * *any odd prime p, bq1;n= 0 for all n 1, where q = 2(p-1). As an application of (b),* * b11 is nilpotent in ExtBif and only if b1nis nilpotent for n 1; as another applica* *tion of (b), if the high powers of t are zero in a quotient B of A_for instance if B* * is finite_then btsis nilpotent when s t. Proof. Part (a) follows from the relation fP0(bits) = bit;s+1. For part (b), we have the following coproduct in B: 2tX i 2t7! p2t-i i: i=0 s0-t Since pi = 0 when i < t, this simplifies to s0-t ps0-t ps0 ps0-t ps0-t p2t 7! 2t 1 + t t + 1 2t : s0-t ps Also since pi = 0 when i < t, then t is primitive in B for every s s0- t. So the above coproduct in B translates to the relation ht;s0-tht;s0= 0 in ExtB. We apply Steenrod operations to this relation: applying fPpt: :f:Pp(fifP1)(fifP0) * *gives t+1 pt bpt;s0= bt;s0-tbt;s0+t If bt;s0+1is nilpotent, then so is bt;s0+t, by part (a). Hence so is bt;s0. * * |___| B.3. NILPOTENCE IN HB**= Ext**B(Fp;Fp) WHEN p IS ODD 113 We have the following conjecture, as a special case of Conjecture B.3.1. To some extent, this would generalize Nakamura's result at the prime 3; on the oth* *er hand, he determines a much smaller nilpotence height than this would. Conjecture B.3.4.Fix an odd prime p. 2(p-1) (a)The element b112 Ext2;2pA (Fp; Fp) is nilpotent. (b)Fix t 1 and let j = p+1_2. Then bttis nilpotent in Ext**B(Fp; Fp), where B = Fp[t; 2t; : :;:jt; jt+1; jt+2; jt+3; : :]:: We have a sketch of a proof (which contains a gap), but it is a bit lengthy,* * and so we relegate it to a subsection. We also include a few other technical result* *s in that subsection. B.3.1. Sketch of proof of Conjecture B.3.4, and other results. Sketch of proof. The proof involves some Massey product manipulations. May's paper [May69 ] is the standard reference for Massey products; many of the key results are reproduced in [Rav86 , A.1.4]. As remarkedsafter Lemma 1.3.10, the element btsis the p-fold Massey product of hts= [pt] with itself. Part (a): In A, we have the coproduct : 2 7-! 2 1 + p1 1+ 1 2: This produces the relation h10h11= 0 in ExtA. Applying the Steenrod operation fifP0 gives the relation h11b11- b0h12= 0: Then for any k 1, we apply fPpk-1: :f:PpfP1to get k pk h1;k+1bp11- b10h1;k+2= 0: Using induction gives the following formula, valid for all k 2: 2+...+pk-2 1+p+p2+...+pk-2 (B.3.5) h11b1+p+p11 = h1;kb10 : So we let N = 1 + 1 + (1 + p) + (1 + p + p2) + . .+.(1 + p + p2+ . .+.pp-2) p-1- 1 = pp______(p:- 1)2 and we look at bN11. Recall that b11 is the Massey product . This p Massey product has no indeterminacy. Lemma B.3.6.The element bN11is contained in the p-fold Massey product p-2 : Proof. By definition definition of Massey products, if y = (wi* *th no indeterminacy), then for any elements x1, : :,:xn, we have yx1: :x:n= x1: :x:n 2 : So we apply this to bN11= bN-111. |__* *_| 114 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k) By our computations with Steenrod operations (i.e., equation (B.3.5)), we ha* *ve p-2 p-2 = : Lemma B.3.7.The Massey product p-2 contains bN-110= 0. We let in = O(n), where O: A -! A is the canonical anti-automorphism. Hence i1 = -1, and Xn i (in) = ii ipn-i: i=0 Proof. We have to prove two things: that the Massey product contains bN-110, and that = 0. To pro* *ve the first of these, one follows the proof of Lemma B.3.6. To prove the second, we show th* *at for all n 2, the n-fold Massey product = 0; this * *goes by induction on n. Once we know this, then applying the Steenrod operation (fP0)i(* *to either the Massey product or to the proof) gives = 0 for any i. Indeed, we show that for each n, the cobar element d[in] is equal to ; hence this Massey product is cohomologousito zero. We also show that this Massey product has no indeterminacy. (Hence d[ipn] kills .) When n = 2, the coproduct i2 7! i1 ip1= 1 p1 gives have the relation h10h11= 0. This starts the induction. When n = 3, we have 2 p i3 7-! i2 ip1+ i1 i2: Since d[i2] = h10h11, then d[ip2] = h11h12, so we have = [i2]h12+ h10[ip2] = d[i3] = 0: nq Since h1n 2 Ext1;pA(Fp; Fp), then the indeterminacy of this Massey product is of the form h10x + yh12; 2) 1;q(1+p) where x 2 Ext1;q(p+pA(Fp; Fp) and y 2 ExtA (Fp; Fp). Since we know that Ext1;*is in one-to-one correspondence with the primitives of A, then Ext1;iis nonzero only when i is of the form pjq for some j. Hence both x and y must be zero, and there is no indeterminacy. * * j Now, we assume that = 0 via ii, for all i < n. Then * *ipi kills for all i < n and all j, and so n-1X i = [ii|ipn-i]: i=1 B.3. NILPOTENCE IN HB**= Ext**B(Fp;Fp) WHEN p IS ODD 115 Hence the coproduct on in shows that this is zero. There is no indeterminacy_for the same reason as when n = 3. |__| So the Massey product in Lemma B.3.6 contains both bN11and 0; hence b11is an element of the indeterminacy. Therefore we need some information about that indeterminacy. The indeterminacy of a p-fold Massey product is contained in the union of certain (p - 1)-fold matric Massey products (see [May69 , 2.3]); if ev* *ery entry in the p-fold Massey product has odd total degree, the same is true of ea* *ch entry in each matrix in the shorter Massey products. Here is a general conjecture about "short" Massey products at an odd charac- teristic. This is the gap in our proof. Conjecture B.3.8.Fix an odd prime p, and fix n < p. Consider an n-fold matric Massey product , in which each entry of each matrix Vihas o* *dd total degree. Then any element of this matric Massey product is nilpotent. (As in [Rav86 , A.1.4], whenever we discuss matric Massey products, we as- sume that the matrices involved have entries with compatible degrees, so that t* *heir products have homogeneous degrees, etc. See [May69 , 1.1] for details.) The conjecture is trivially true when n < 3, by graded commutativity; in particular, it is true when p = 3. Indeed, when p = 3, we see that every elemen* *t of the indeterminacy is nilpotent of height p. One can also show that the conjectu* *re is true when taking the Massey product of one-dimensional classes in the cohomology of a space [Dwy ]; otherwise, we do not have much evidence for it. Meanwhile, it has the following consequence. Conjecture B.3.9.In particular, if n < p and if ai has odd total degree for i = 1; : :;:n, then for any element x contained in is nilpotent. Since bN11is an element of the indeterminacy of a p-fold Massey product of odd-dimensional classes, we may conclude that bN11is nilpotent. This would fini* *sh the proof of Conjecture B.3.4(a). Part (b) of the conjecture would be proved similarly. One starts with the relation ht0htt= 0 in ExtB and applies Steenrod operations to get the following replacement for (B.3.5): t-1+p2t-1+...+pkt-1 pt-1+p2t-1+...+pkt-1 httbptt = ht;(k+1)tbt0 : If we set M = 1 + pt-1+ (pt-1+ p2t-1) + . .+.(pt-1+ p2t-1+ . .+.p(p-1)t-1); then we get bMtt= bM-1tt t-1 pt-1+p2t-1 pt-1+...+p(p-1)t-1 2 t-1 pt-1+p2t-1 pt-1+...+p(p-1)t-1 = 3 bM-1t0 = 0: So Conjecture B.3.9 implies that bMttis nilpotent. |___| As remarked above, the gap in the proof_Conjecture B.3.8_is not a gap when p = 3, so Conjecture B.3.4 holds at the prime 3: 116 B. STEENROD OPERATIONS AND NILPOTENCE IN Ext**(k;k) Proposition B.3.10.Fix p = 3. (a)The element b112 Ext2;36A(F3; F3) is nilpotent. Indeed, b1511= 0. (b)Fix t 1. Then bttis nilpotent in Ext**B(F3; F3), where B = F3[t; 2t; 2t+1; 2t+2; 2t+3; 2t+4; : :]:: Arguing similarly, we see: Proposition B.3.11.Fix p = 3. The element b222 Ext2;432B(F3; F3) is nilpo- tent, where B = A=(1). 2 Proof. The coproduct 4 7! p2 2 gives the relation h20h22= 0 in ExtB. Hence h2;ih2;i+2= 0 for all i. As in the "proof" of Conjecture B.3.4, we find that b1622is contained in the Massey product ; this Massey product also contains the* * ele- ment b1520. We claim that the three-fold Massey product is nilpotent. i This Massey product is defined because the product h2;ih2;i+2is killed by 34. So we consider the diagonal on 6: 2 33 34 6 7! 34+ 3 3+ 2 4 = + h30h33: We have used the fact that 3i3is primitive in A=(1), and hence gives rise to a 1-dimensional Ext class h1;i. Hence we have h30h33= -: By graded commutativity, (h30h33)2 = 0; hence the same is true of the Massey product. We find that b1622= b1510+ x for some class x in the indeterm* *inacy of a three-fold Massey product. Both x and are nilpotent,_and h* *ence so is b22. |__| Corollary B.3.12.Fix p = 3. 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Soc. 73* * (1967), 647- 648. 120 BIBLIOGRAPHY Index A, 23 complex act nilpotently, see nilpotent actioncobar, 6 Adams spectral sequence, see spectralcse-omultiplication, 2 quence, Adams conjugation, 2, 69 Adams tower, 104 connective, 14, 18 admissible, 64, 84-86 weakly, 14 A(n), 26, 35, 39, 42, 94, 96 conormal, 16, see Hopf algebra, quotient, conormal bicommutative, 3 coproduct, 7 Bousfield class, 18, 57, 73, 81 cotensor product, 8 Bousfield equivalence, 18, 57, 73 BP, 48 D, 48, 51 fifP0, 11, 60 degree, 3, 11 fifPn, 109 derived category, see category, derived Brown category, 7 detect nilpotence, 50, 51, 73 Brown-Comenetz dual, see dual, Brown-ComenetzD(n), 51, 67 bts, 25, 40, 42, 43, 61, 111 Dr, 57 nilpotence of, 60, 111-116 Dr;q, 57 dual category Brown-Comenetz, 18, 73 derived, 7 Spanier-Whitehead, 33, 65, 105 model, 101-102 DX, see dual, Spanier-Whitehead cofibrantly generated, 102 D[x], 11 stable homotopy, 6 D[ps], 25, 40 connective, 13 t cellular tower, 13 E-complete, 107 Chouinard's theorem, 20, 61 elementary, see Hopf algebra, elementary chromatic convergence, 97-99 E[on], 40 chromatic tower, 97, 98 E[on], 25 CL, 14, 34, 42, 90 E[x], 11 closed model category, see category,Emodelxt, 6, 7 coaction, 4, 11, 48, 58, 69, 71 ExtA, 34 diagonal, 4 extension, see Hopf algebra extension; group coalgebra, 2 extension quotient, 8-10, 43 cobar complex, see complex, cobar F-isomorphism, 48, 49, 59, 64, 69, 78 cocommutative, 3 uniform, 48 cofiber sequence, 6, 8, 12, 19 fibration, 102 cofibration, 102, see also cofiber sequencefield spectrum, see spectrum, field cofree, 5, 94 finite spectrum, see spectrum, finite cohomology functor, see functor, cohomol-finitejtype,118jm ogy F(ud1; : :;:udm), 80-84, 88-92 coideal F(Un), see F(uj1d1; : :;:ujmdm) coaugmentation, 11 functor comodule, 4 cohomology, 8 simple, 6, 13, 19, 101 exact, 94 comodule-like, see CL homology, 8 121 122 INDEX 2Bk, 11 Margolis' killing construction, 87, 90 ==C, 11 Massey product, 12, 110, 111, 113-116 generated, 6 M, 92 generic, 15, 17, 53, 65, 103, 107 Mnf, 97, 98 group algebra, 19 monogenic, 7, 13, 18 group extension, 19 Morava K-theories, 42, 65 grouplike, 4 morphism, 6, 94 grading, 7 H, 8-12, 35 stable, 94 Tate version of, 94-97 bH, see H, Tate version of nilpotence theorem, 50 HD, 48, 50, 51, 59, 65, 69, 83, 92 nilpotent action, 36, 41 Hom_, 94 Nishida's theorem, 49 homology functor, see functor, homologynormal, see Hopf algebra, sub-, normal homotopy group, 8 Hopf algebra, 2, 101 fP, 109 connected, 3, 6, 14 P-algebra, 26 elementary, 27, 29, 38, 41 periodicity theorem, 64, 65, 76, 92 quasi-elementary, 29, 48-50, 58, 73ss**, 8, 34 quotient, 8, 9, 24, 36, 48, 73 PM, 48 conormal, 11, 16, 24, 48, 51, 58,P73ostnikov tower, 13 maximal elementary, 27 primitive, 4, 6, 11, 43, 48, 60, 102, 109, 1* *11, maximal quasi-elementary, 57, 69, 71112 sub- profile function, 24 normal, 11, 74 projective, 5 Hopf algebra extension, 10, 19, 25,p36,r37,ojectivesclass, 107 40, 41, 51, 73 pt, 30,s84 horizontal vanishing line, see vanishingPline,t,s29, 30, 38, 42, 79 horizontal Pt-homology, 30 hts, 25, 36, 42, 43, 54, 69, 71, 110,m111odule,s30, 31 nilpotence of, 36, 54, 57, 60, 110Pt-homology spectrum, 30, 31, 80 Hurewicz map, 14, 42, 49, 51 Q, 49, 58, 69 Hurewicz theorem, 14 __Q, 58 ideal, 63, 76-78 qn, 30 augmentation, 11 Qn, 29, 30, 42, 80 invariant, 76, 78, 83, 92 Qn-homology, 30 induction, 94 Qn-homology spectrum, 30 injective, 5, 6, 9-11, 30, 43, 94 quasi-elementary, see Hopf algebra, quasi- intercept, 35 elementary invariants, 48, 69 Quillen stratification, 49, 61 IX, see dual, Brown-Comenetz rank variety, see variety, rank I(X), see ideal relative vanishing line, see vanishing line, kG, 19 relative resolution LfC, see localization, finite injective, 6, 101 Lfn, see localization, finite restriction, 9, 94 Lfn, 95 restriction map, 9, 43, 51 LfnA(m), 96 ring spectrum, see spectrum, ring loc(A), 75, 92 self-map, 42, 50 loc(), 15 central, 44, 64 loc(Y ), 15 Shapiro's lemma, 10 localization shearing isomorphism, 10 Bousfield, 88, 92 i;j, see suspension finite, 87, 88, 94, 97 slope, 31, 34, 41, 42, 64, 80, 84-86, 88, 95, smashing, 88, 93 97 localizing subcategory, see subcategory,slo-lope support, 64, 84-86 calizing Slopes, 64, 84-86 INDEX 123 Slopes0, 64, 84-86 |x|, see degree small, 15 X-acyclic, 18 smash product, 7 n, 23 Spanier-Whitehead dual, see dual, Spanier- Whitehead [y], 11, 60 spectral sequence y-element, see y-map Adams, 16, 17, 52, 102-108 y-map, 64-67, 83 construction, 103-104 strong, 65, 66 convergence, 107 yn, 32, 43, 95 Atiyah-Hirzebruch, 42, 77 Z(n), 31, 34, 42, 64, 80-83, 88, 90, 97, 98 change-of-rings, see spectral sequence,zex-(n), 32, 42, 43 tension in, 69 extension, 16, 51, 52 Lyndon-Hochschild-Serre, 16 spectrum, 6 field, 10, 65 finite, 15, 50, 64, 75, 76, 78, 80, 83, 84, 87, 89, 98 ring, 8, 9, 18, 50, 65, 66, 98 sphere, 7 Sqn, 71 fSqn, 109, 110 stable homotopy category, see category, sta- ble homotopy Stable(A), 24 Stable(), 6, 101 StComod(B), 94 Steenrod algebra, 23 Steenrod operation, 11, 17, 109-110 subcategory localizing, 15, 34, 88 thick, 15, 54, 65, 78-79, 81, 83, 87, 94, 103 suspension, 7, 94 telescope, 18, 83, 88 telescope conjecture, 92 thick subcategory, see subcategory, thick thick(Y ), 15 on, 23 trivial comodule, 13 type n, 42, 83, 89 un, 42, 43, 64 un-map, 42, 80-83, 88 unit map, 8 vanishing line, 34-44, 92, 107-108 horizontal, 34, 38, 41 relative, 42 vanishing plane, 17, 53, 102-107 variety rank, 79-80 vn, 25, 40, 42, 43, 111 vn-map, 42, 65 weak equivalence, 102 Wj, 53, 54 , see Bousfield class [x], 11, 109